VNR Concise Encyclopedia of Mathematics 9789401169820, 9401169829

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VNR Concise Encyclopedia of Mathematics
 9789401169820, 9401169829

Table of contents :
Content: Preface
Authors and translators
I. Elementary Mathematics
II. Steps towards higher mathematics
III. Brief reports on selected topics.

Citation preview




CONCISE ENCYCLOPEDIA OF MATHEMATICS ~~~~~~ WGellert· S.Gottwald M. Hellwich . H. Kastner· H. KOstner Editors KA.Hirsch· H.Reichardt Scientific Advisors


_ _ _ _ New York

© VEB Bibliographisches Institut Leipzig,


Softrover repri1t of the hardoover 1st edition 1975

Mathematics at a Glance First American Edition 1977 Second American Edition 1989 Library of Congress Catalog Card Number 88-26992 1SBN-13: 978-94-011-69844 e-lSBN-13: 978-94-011-6982-0 001: 10.1007!978-94-011-6982-O All rights reserved. No part of this work covered by the copyright hereon may be reproduced or used in any form or by any means graphic, electronic, or mechanical, including photocopying, recording, taping, or information storage and retrieval systems without written permission of the publisher. Made in the German Democratic Republic. Published by Van Nostrand Reinhold 115 Fifth Avenue New York, New York 10003 Van Nostrand Reinhold International Company Limited 11 New Fetter Lane London EC4P 4EE, England Van Nostrand Reinhold 480 La Trobe Street Melbourne, Victoria 3000, Australia Macmillan of Canada Division of Canada Publishing Corporation 164 Commander Boulevard Agincourt, Ontario MIS 3C7, Canada 16















Library of Congress Cataloglng-In-Publlcation Data Main entry under title: The VNR concise encyclopedia of mathematics. First published under title: Mathematics at a glance. Includes index. I. Mathematics-Handbooks, manuals, etc. I. Gottwald, S. II. Van Nostrand Reinhold Company. QA40.v18






Introduction ..................•.....•...•..•.••.....•.......••.........•......... I. Elementary mathematics 1. Fundamental operations on rational numbers. . . . • . . . . . . . . . . . . . . . . . . . . . . . . . • . . . • 2. Higher arithmetical operations............................... .•...... ........• 3. Development of the number system............................................ 4. Algebraic equations ............•...•..•...•..•.........•...................• 5. Functions ................................................................. , 6. Percentages, interest and annuities ...•..............•..........•...........•.. , 7. Plane geometry .........•.............•••............•••••.•...•...•.......• 8. Solid geometry .................•...........•..••.......•....••.......•.•.•. , 9. Descriptive geometry ............•••••.•.•..............•.........•.....•...• 10. Trigonometry ...............•.......................................••..•... 11. Plane trigonometry ............•..........•......•.....•.........•...•...... , 12. Spherical trigonometry .....................................•.•..........•..• 13. Analytic geometry of the plane ...•.••....•.....•...•.......••••..••..••....... ll. Steps towards higher mathematics 14. Set theory ..................••.....••.......••..•.........••.•........•..... 15. The elements of mathematical logic ...............••................•..•...... 16. Groups and fields ...............••.....•........•.......................... 17. Linear algebra . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . . .. 18. Sequences, series, limits. . . . . . . . . . . . . • • . . . • • . • . . • . . • . . . . . • . . • . . . . • • . . • • . . . • . .. 19. Differential calculus .•.........•............................................. 20. Integral calculus .•.........................•.....•.....•.......•••••.••..... 21. Series of functions. . . . . . . . . . • . . . . . • . . . • . . . • . . • . • . . . . . . . . • . . • . . . . . . • • . • . . . . . .• 22. Ordinary differential equations. . . • • . . . . . . . . . . • . . . • • . . . . . . . . • • . • • • . . . . • • • • • . . .. 23. Complex analysis .................•.....•••••••......•...•.•...•..••.......• 24. Analytic geometry of space ..........•..•.................•..........••...... 25. Projective geometry ......••..•...........•...••..•....••..•.•..•..••.•.•..•. 26. Differential geometry, convex bodies, integral geometry •....••.•.••.•..•..•...... 27. Probability theory and statistics ..•....••.......•..•..............•.•...•..... 28. Calculus of errors, adjustment of data, approximation theory . • • . . . • . . . . . . . . . . • . •. 29. Numerical analysis •...••...........•...••.••.•.............••.....•......... 30. Mathematical optimization. • . . . • . . . • . . . . . . . . . . . . . . . . . . . . . • . . . . . • . . . . . . . . . . . .. m. Brief reports on selected topics 31. Number theory •.•...............•••••.••............•...•....••.......•...• 32. Algebraic geometry .•....•.•...........•...••••••.........•••............... 33. Further algebraic structures ..••...••......•••..•.•.................•......•.. 34. Topology .•........•...••.....•.......•...•....••.......................... 35. Measure theory. . . . . . . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . • . • . . . . . . . . . . . . . . . . .. 36. Graph theory. . . . . . . . • . . . . . . . . . • • . • • . . . . . • • . . • • . . . . . . . . . . . . . . . . . . . . . . • . . . . .• 37. Potential theory and partial differential equations ..•.....•.......•.............. 38. Calculus of variations. . . . • . • • • • . . . • • • . . . . . . . . . . . • . . . . . • . . . . . • . • • . • . . . . . . . . . .. 39. Integral equations .......•..•••....••......................•....•........... 40. Functional analysis. . . . . . . . . . . . . • . . . . . • . . . • . . . • . . . . . . . . . . . . . . • . . . . . . . . . . . . . .. 41. Foundation of geometry - Euclidean and non-Euclidean geometry ................ 42. Foundations of mathematics .............•................................... 43. Game theory ......•........................................................ 44. Perturbation theory ......................................................... 45. The pocket calculator .•............• . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 46. Microcomputers .....•.............•........................................

11 17 47 69 80 107 139 146 184 203 220 241 261 282 320 332 343 356 381 406 443 479 500 517 530 547 561 575 607 630 653

669 675 678 680 687 688 693 698 703 705 711 717 723 731 732 745


It is commonplace that in our time sc:iem:e and technology cannot be mastered without the tools of mathematics; but the same applies to an ever growing extent to many domains of everyday life, not least owing to the spread of cybernetic methods and arguments. As a consequence, there is a wide demand for a survey of the results of mathematics. for an unconventional approach that would also make it possible to fill gaps in one's knowledge. We do not think that a mere juxtaposition of theorems or a collection of formulae would be suitable for this purpose, because this would over· emphasize the symbolic language of signs and letters rather than the mathematical idea, the only thing that really matters. Our task was to describe mathematical interrelations as briefly and precisely as possible. In view of the overwhelming amount of material it goes without saying that we did not just compile details from the numerous text-books for individual branches: what we were aiming at is to smooth out the access to the specialist literature for as many readers as possible. Since well over 700000 copies of the German edition of this book have been sold, we hope to have achieved our difficult goal. Colours are used extensively to help the reader. Important definitions and groups of formulae are on a yellow background, examples on blue, and theorems on red. The course of more complic:ated calculations is indicated by red arrows. Also, in the illustrations in the text colours show up the essential features. Ample examples help to make general statements understandable. Frequently the numerical calculations have been arranged separately so that a problem can be read as ali explanatory text, without reference to calculations, while the latter can be regarded as worked exam· pies with explicit details. Physical units, which occur in some examples, are given in the SI-system, which is coming more and more into legal and practical use. Eveayday examples are given in everyday units, both metric and others. A systematic subdivision of the material, many brief section headings, and tables are meant to provide the reader with quick and reliable orientation. The detailed index to the book gives an easy access to specific questions. In the plates at the end numerous photographs and colour plates help to make the material more vivid and provide interesting glimpses of the history of mathematics. We thank the authors of the various chapters, specially to acceding to our request for generally understandable diction even at the risk of deviating from the usual terminology. Above all in the brief reports on special topics many an author has found it difficult to be content with mere in· dications about a topic in which he is an expert. Our particular thanks are due to our advisors, Professor K. A. Hirsch, Queen Mary College, University of London, and Professor H. Reichardt, Section for Mathematics, Humboldt University of Berlin. They have worked untiringly for the improvement of the book and have helped to create a work which is a reliable source of information for every user and should convince everyone that mathematics is essentially a simple and learnable discipline. The Editors and the Publishers


Archimedes Poster of the town of Syracuse (Italy) 2/3 Mathematics in sdtool l/ll Introduction of the number seven and revision exercises 4 Mathematics in industrial arts Surfaces of revolution in the design of a pottery set 5 Drawing instruments I Geometry sets 6 Drawing instruments II Slide rules 7 Drawing instruments ill Rulers, protractors, and French curves 8 Graph papers Millimetre paper - doubly logarithmic paper - simply logarithmic paper - polar coordinate paper - triangular net paper probability paper 9 From the earliest period of mathematics Clay vessels of the new stone age Early Egyptian surveying 10 Ancient Egyptian mathematics Original text of the Hau problem in Demotic writing and transcription of the same text into hieroglyphics Calculation of a frustum of a pyramid 11 Babylonian mathematics Cuneiform tablet with calculations of areas Section of the tablet above 12 Graeco-Roman mathematics The Elements of Euclid, first printed edition 1482 Roman hand abacus 13 Andent Chinese mathematics From a manuscript dated 1303 Bamboo sticks to represent numbers Chinese slide rule (about 1600) 14 Andent Hindu mathematics Mathematical-astronomical buildings of the 17th century Mathematical manuscript of the 16th century 15 Arabic: mathematics Theorem of Pythagoras in an Arabic mathematical manuscript of the 14th century Arabic astrolabe 16 Mathematics in Europe, 15th to 17th century Triumph of the modem algorithm (digital calculation) over the ancient counter reckoning (abacus) The use of Jacob's staff



19 20


Mathematics and the visual arts I Ancient Egyptian mural: catching fish and hunting birds in a papyrus thicket Painting by Melozzo da Forli (1438-1494): Pope Sixtus IV appoints Platina as Prefect of the Vatican Mathematics and the visual arts II Proportions of the human body Drawings by Leonardo da Vinci and sketch by Albrecht DOrer Mathematics and the visual arts ill Melancholia, copper engraving by Albrecht Viirer Geometric: forms in architecture and technology I Egyptian pyramids near Giza Tower of city walls The old town hall of Leipzig Geometric forms in architecture and tecbnologyll

Modern water tower Cooling towers of a generating plant 22 Geometric: forms in architecture and technologyill






Obelisk in the great temple of Amun at Karnak Wedge as a cleaving tool Hyperbolic paraboloid shells as roofs of an exhibition hall Famous mathematicians of the 15th/16th century Regiomontanus - Simon Stevin - Albrecht Diirer - Niccolo Tartaglia - Geronimo Cardano - Jost Biirgi - Luca Pacioli Famous mathematicians of the 16th century Title page of Robert Recorde's ' Algebra' Title page of Adam Ries's 'Rechnung auff der Linihen und Federn ... ' A problem out of this book concerning the purchase of livestock From old arithmetic books Conclusion of a business deal at a calculating desk Calculation of the capacity of a cask Two libraries The mathematics room of the National and University Library in Prague Entrance to the Science Library of Erfurt (Boyneburg portal) Old matbematkal aids I Pedometer, 1741 Slit bamboo as counting stick Tally stick

8 Old ..........ticaI aids II Counters or markers for arithmetic and an elaborate box, 16th century 29 Old matbelllatical aids m Surveyor's compass, about 1600 30 0 ........ 1 Illustration of a rod, by juxtaposition of 16 feet 16th century measuring rods with various graduations 31 01d . . . . . . 11 Set of weights, Nuremberg 1S88 Hinged sun dial, ivory 32 F~ matbelllatidaDs of tile 17th ceaDryl Title page of Descartes' 'Discours de Ja mabode' Ren6Descartes 33 FIUIIOUS matbematiclans of tile 17th ceaturyll Fran~is Vieta - John Napier - Galileo Galilei - Johannes Kepler - Buonaventura Cavalieri - Pierre de Fermat - James Gregory 34 F~ matbematidaDs of tile 17th/18th ceatury 1 Blaise Pascal Gottfried Wilhelm Leibniz Isaac Newton 35 FIUIIOUS matliematiciaDs of tile 17th/18th eent.yll Extract from a manuscript of Leibniz with the integral sign The mechanical calculator constructed by Pascal in 1642 36 F~ matbematidans of tile 17th/18th centurym Jakob Bernoulli Johann Bernoulli Daniel Bernoulli 37 Famous matbematiclans of the 18th ceaturyl Page from a manuscript by Euler Leonhard Euler 38 FaDlOlIS mathematicians of the 18th ceaturyll Brook Taylor - Moreau Maupertuis Johann Heinrich Lambert - Joseph Louis Lagrange - Gaspard Monge - Adrien Marie Legendre - Jean Baptiste Joseph de Fourier 39 FaDlOlIS mathematidaDs of the 1M ceaturyl Drawing by Janos B6lyai on non-Euclidean geometry Nikolai lvanovich Lobachevskii 40 Famous mathematicians of the 19th ceaturyll Portrait of the young Gauss Gauss in his old age Gauss's signature The University in Gottingen 41 Famous matbematicians of the 19th ceatury III A page from Gauss's scientific diary 28














matbelllatfdus of tile 1M

c:eDtIIrJ IV

Friedrich Wilhelm Bessel - Augustin Louis Cauchy - Jakob Steiner - Niels Henrik Abel - Peter Gustav Lejeune Dirichlet :avariste Galois - Pafnuti Lvovich ChebysheY F~ matbelllatfdus of tile 1M ceatIIr)' V Carl Gustav Jacob Jacobi - Bernhard Riemann - Leopold Kronecker - Karl WeierstraB - Arthur Cayley - Sophus LieSonya KovaJevskay Matbelllatical iIIstruDalts 1 Instrument for drawing an integra) curve of a given function or differential equation Instrument to evaluate the integra) of a function whose graph is given Mathematical instrumeats II Compensating polar planimeter with polar arm Compensating polar planimeter with polar carriage Mathematical instrumeats m Precision pantograph Instrument for the measurement of reelangular coordinates or the drawing of points with given coordinates Matbematical instruments IV Harmonic analyser Instrument to determine the tangent or normal to a curve whose graph is given FamollS matllematidans of the 1M/20th century 1 George Stokes - Richard Dedekind - Georg Frobenius - Georg Cantor - Henri Poincare - Felix Klein - Emmy Noether Famous matbematicians of the 19th/20th century II David Hilbert - :alie Joseph Cartan - Henri L60n Lebesgue - John von Neumann Hermann Weyl - Jacques Hadamard Stefan Banach Surveying Signals for the observation of trigonometric nets Trigonometric point (TP) Matbematical education I Work on a wall board Determination of an angle with a hand-made apparatus Giant slide rule for instructional purposes. Matbematical education II Computations on part of an exhaust system Geometrical constructions on the blackboard Application of Pythagoras' theorem Mathematical education m Models for pupils: Cube with surface and space diagonals - Prism decomposable into three pyramids of equal volume - Cylinder with sections - Sphere with plane sections Sections of a right circular cone

9 S4


Mirror images Negative and positive of a photograph Reflection in water Ship's Diesel engine in a left- and righthand version Variational problems Formation of a minimal surface in a lobster pot Formation of a minimal surface by a soap film

The patli of-thelight ray is the solution of a minimal problem S6 Mathematical models Moebius strip A closed surface of genus 1 Pseudosphere Surface representing the modulus of the function w = exp (l/z)

Index of mathematicians Abel, Niels Henrik, 1802-1829 d'Alembert, Jean Ie Rond, 1717-1783 Apollonius of Perga, c. 262-190 ? B. C. Archimedes, 287?-212 B. C. Argand, Jean Robert 1768-1832 Aristotle, 384-322 B. C. Banach, Stefan, 1892-1945 Beltrami, Eugenio, 1835-1900 Bernoulli, Daniel, 1700-1782 Bernoulli, Jakob, 1654-1705 Bernoulli, Johann, 1667-1748 Bessel, Friedrich Wilhelm, 1784-1846 Bezout, Etienne, 1730-1783 Bhaskara, 1114-1185? Birkhoff, George David, 1884-1944 Blaschke, Wilhelm, 1885-1962 B6lyai, Farkas, 1775-1856 B6lyai, Janos, 1802-1860 Bolzano, Bernard, 1781-1848 Bombelli, Rafael, 16. century Bahmagupta, born 598 Briggs, Henry, 1561-1630 Brouwer, Luitzen Egbertus Jan, 1881-1966 Buffon, Georges Louis de, 1707-1788 Burgi, Jost, 1552-1632 Burnside, William, 1852-1927 Cantor, Georg, 1845-1918 Caratheodory, Constantin, 1873-1950 Cardano, Geronimo, 1501-1576 Cartan, Elie Joseph, 1869-1951 Cartesius t Descartes Cauchy, Augustin Louis, 1789-1857

Cavalieri, Bonaventura, c. 1598-1647 Cayley, Arthur, 1821-1895 Ceva, Giovanni, 1647-1734 Chebyshev, Pafnuti Lvovich, 1821-1894 Clavius, Christoph, 1537-1612 Cramer, Gabriel, 1704-1752 Cusanus, Nicolaus, 1401-1464 Dandelin, Pierre, 1794-1847 Dedekind, Richard, 1831-1916 de la Vallee-Poussin, Charles, 1966-1962 Descartes, Rene, 1596-1650 Diphantos of Alexandria, c. 250 A. D. Dirichlet, Peter Gustav Lejeune, IW5-1859 Durer, Albrecht, 1471-1528 Eisenhart, Luther Pfahler, 1876-1965 Enriques, Federigo, 1871-1946 Eratosthenes of Kyrene, c. 276-194 B. C. Euclid of Alexandria, c. 450-380 B. C. Eudoxus, c. 408-355 B. C. Euler, Leonhard, 1707-1783 Fermat, Pierre de, 1601-1665 Ferrari, Ludovico, 1522-1565 Ferro, Scipione del, c. 1465-1526 Fibonacci t Leonardo of Pisa Fisher, Ronald Aylmer, 1890-1962 Fourier, Jean Baptiste Joseph de, 1768-1830 Fraenkel, Abraham, 1891-1965 Fredholm, Erik Ivar, 1866-1927 Frege, Gottlob, 1848-1925 Frobenius, Ferdinand Georg, 18491917 Galilei, Galiko, 1564-1642


Index of mathematicians

Galois, Evariste, 1811-1832 GauB, Carl Friedrich, 1777-1855 Girard, Albert, 1595-1632 Goldbach, Christian, 1690-1764 Green, George, 1793-1841 Gregory, James, 1638-1675 Guldin, Paul, 1577-1643 Gunter, Edmund, 1561-1626 Hadamard, Jaques Salomon, 1865-1963 Hamilton, Sir William Rowan, 1805-1865 Hankel, Hermann, 1839-1874 Herbrand, Jacques, 1908-1931 Hermite, Charles, 1822-1901 Heron of Alexandria, c. 75 A. D. Hesse, Ludwig Otto, 1811-1874 Hilbert, David, 1862-1943 Hippasos of Metapontum, c. 450 B. C. Hippocrates of Chios, c. 440. B. C. I'Hospital, Guillaume Fran~ois Antoine Marquis de, 1661-1704 I'Huilier, Simon, 1750-1840 Huygens, Christiaan, 1629-1695 Jacobi, Carl Gustav Jacob, 1804-1851 Jordan, Marie Ennemond Camille, 1838-1922 Kepler, Johannes, 1571-1630 Klein, Felix, 1849-1925 Kovalevski, Sonya, 1850-1891 Kronecker, Leopold, 1823-1891 Krull, Wolfgang Adolf ludwig Helmuth, 1899-1971 Kummer, Ernst Eduard, 1810-1893 lagrange, Joseph Louis, 1736-1813 Lambert, Johann Heinrich, 1728-1777 Laplace, Pierre Simon de, 1749-1827 Lasker, Emmanuel, 1868-1941 Lebesgue, Henri Leon, 1875-1941 Legendre, Adrien Marie, 1752-1833 Leibniz, Gottfried Wilhelm, 1646-1716 Leonardo da Vinci, 1452-1519 Leonardo of Pisa, called Fibonacci, 1180?-1250? Lie, Sophus, 1842-1899 Lindemann, Ferdinand von, 1852-1939 Liouville, Joseph, 1809-1882 Lipschitz, Rudolf, 1832-1903 lobachevskii, Nikolai Iwanowich, 1792-1856 Lullus, Raimundus, Lull, Ramon, c. 1235-1315 Machin, John, 1685-1751 Maclaurin, Colin, 1698-1746 Maupertuis, Pierre Louis Moreau de, 1698-1759 Menelaus of Alexandria, c. 98 A. D. Miilkowski, Hermann, 1864-1909 Mobius, August Ferdinand, 1790-1868 Moivre, Abraham de, 1667-1754 Monge, Gaspard, 1746-1818 Morgan, Augustus de, 1806-1871 Napier, Neper, John, 1550-1617 Neumann, John von, 1903-1957 Newton, Isaac, 1643-1727 Noether, Emmy, 1882-1935 Noether, Max, 1844-1921 Oresme, Nicole, 1323?-1382

Ostrogradskii, Michail Wassilyevich,1801-1862 Ought red, William, 1574-1660 Pacioli, luca, 1445?-1514 Partridge, Seth, 1603-1686 Pappus of Alexandria, 4. century Pascal, Blaise, 1623-1662 Peano, Giuseppe, 1858-1932 Pearson, Karl, 1857-1936 Pell, John, 1610-1685 Plato, 427-347? B. C. Plucker, Julius, 1801-1868 Poincare, Henri, 1854-1912 Poisson, Simeon Denis, 1781-1840 Poncelet, Jean Victor, 1788-1867 Poseidon ius, c. 135-51 B. C. Proclus,c.410-485 Pythagoras of Samos, c. 580-496 B. C. Quetelet, Lambert Adolphe Jacques, 1796-1874 Recorde, Robert, 1510?-1558 Regiomontanus, Johannes, 1436-1476 Riemann, Bernhard, 1826-1866 Ries, Adam, 1492-1559 Rolle, Michel, 1652-1719 RudoltT, Christoph, c. 1500-1545 Ruffini, Paolo, 1765-1822 Russell, Bertrand, 1872-1970 Rytz, David, 1801-1868 Saccheri, Girolarr.o, 1667-1733 Schmidt, Erhard, 1876-1959 Schwarz, Hermann Amandus, 1843-1921 Segre, Corrado, 1863-1924 Severi, Francesco, 1879-1961 Simpson, Thomas, 1710-1761 Staudt, Carl Georg Christian von, 1798-1867 Steiner, Jakob, 1796-1863 Stevin, Simon, 1548-1620 Stifel, Michael, 1487-1567 Stirling, James, 1696-1770 Stokes, George Gabriel, 1819-1903 Tartaglia, Niccolb, originally Fontana Niccolb, c. 1500-1557 Taylor, Brook, 1685-1731 Thales of Miletus, c. 624-547 B. C. Theaitetus, 410?-368 B. C. Theodoros von Cyrene, c. 390 B. C. Tschirnhaus, Ehrenfried Walter Graf von, 1651-1708 Viet a t Viete Viete, Fran~ois, 1540-1603 Vlacq, Adrien, c. 1600-1667 Wallis, John, 1616-1703 Waring, Edward, 1734-1798 WeierstraB, Karl, 1815-1897 Wessel, Caspar, 1745-1818 Weyl, Hermann, 1885-1955 Whitehead, Alfred North, 1861-1947 Widmann, Johann, born 1460 Wingate, Edmund, 1593-1656 Wittich, Paul, 1555-1587 Wronski, Josef Maria, 1775-1853 Zenodoros, c. 180 B. C. Zenon of Elea, 490-430 B. C. Zermelo, Ernst, 1871-1953


The great achievements of technology in all its forms, which deeply influence the life of every human being, have led to a widespread recognition of the importance of mathematics: everybody knows, or at least believes, that without mathematics these achievements in their entirety could not have come about. Interest in mathematics has therefore grown steadily, and with it the need for information about this science. Now in many respects mathematics is an exceptional science, in particular, as regards the presentation of its problems and results. While in medicine, zoology, botany, geography and geology, or in languages, history, astronomy, a scholar, fully equipped with the knowledge of his time, can explain to a layman the majority of his problems and results, perhaps even his methods or the fundamental principles of his special interests, in such a way that he succeeds in conveying an impression of the contents of this field, in present-day chemistry and physics this is far more difficult - and in mathematics well-nigh impossible. Not only has the volume of results grown phenomenally, but the problems are so difficult to treat and lie so deep that even mathematicians can have no more than a superficial view of the whole of mathematics. One tries to counteract the fragmentation of mathematics into many special branches by extracting as far as possible from various domains common features, which sometimes do not lie at all close to the surface, and by creating from them a new and even more abstract theory: in just this way new links are forged between at first sight widely diverging directions. This process can be regarded as a repeated abstraction: whereas the basic disciplines such as algebra and geometry have their origin in abstractions from everyday experience, one arrives at such a unifying theory by further abstractions, for example, from algebra and geometry: and under certain circumstances such abstracting processes can be repeatedly piled on top of one another. Here 'abstract' has to be understood in the literal meaning of the word as 'removing', as leaving aside everything inessential for the context in question or for a particular purpose; for example, ignoring colour in geometric figures, which may very well playa role in ornaments. From all this it follows that it is quite impossible to give a layman even a glimpse of the whole of contemporary mathematics. Here a layman is not only one whose knowledge is limited to the normal contents of a school syllabus. Even a mathematician with a diploma or a B. Sc., even a teacher of mathematics, has to be regarded as a layman in many special branches. It is simply impossible to acquire specialized knowledge of all branches of mathematics in three or four years of study. Therefore this book cannot have the ambition of imparting knowledge in all special fields of mathematics - restriction is essential. In its historical development mathematics first proceeded in quite a naive manner. It started out from the numbers 1, 2, 3, ... and from the intuitively obvious figures of geometry such as points, segments, lines, planes in space, angles, triangles, circles, etc.; gradually it ascended to more complex formations, with the realm of numbers and that of figures not developing as separate entities, but connected through the notion of measuring. It was in this development, progressing from the intuitively simple and obvious to more complicated problems, that mathematics was built up, for example, in Babylonia and Egypt; astonishing achievements were reached in astronomy, such as the prediction of lunar eclipses. But it was the Greeks who lifted mathematics to a completely new level of development when they felt compelled not always to forge ahead, but also to reflect: what is it that one does in pursuing mathematics? The result was that through them mathematics became a sience in the present-day sense. On the one hand, they recognized that a proof consists in reducing a mathematical proposition to other known facts by the simplest logical conclusions, supported and made convincing sufficiently often by evidence or experience. On the other hand, they realized that such a reduction process cannot go on indefinitely but only as far as certain simplest properties of numbers or figures, which appear secure by virtue of intuition or experience. In this way they compiled for the first time consciously a system of fundamental facts, for example, that there is precisely one straight line passing through two points, and they created the foundation of logic. Together these two features lead to a systematic build-up of geometry, rising from the simple to the complex. For a long time this Euclidean geometry, apart from a few minor supplements, remained the model of a science. However, no comparable attempt was made for about two thousand years to

12 treat algebra and later analysis in the same manner. The basic properties of the natural numbers were something obvious for the Greeks, but questions of divisibility and problems concerning prime numbers were of interest to them. They knew how to manipulate common fractions, but they did not pursue the idea of introducing negative numbers. However, in connection with a right-angled isosceles triangle they stumbled on the fact that fractions are insufficient to describe the ratios of all quantities: they noticed that in such a triangle the ratio of side to base cannot be represented by a fraction. But from this they did not by any means draw the conclusion that the domain of fractions ought to be extended in such a way that this ratio, and as far as possible all other geometric ratios, could be described numerically in terms of the new numbers of the more extensive domain. They did precisely the opposite: they geometrized their algebra. True, this led to a theory that is equivalent to our theory of real numbers; but the geometrization gave rise to such complications that Greek mathematics ground to a halt. Centuries later the practical needs of astronomers and mariners required urgently trigonometric calculations, which could only be mastered with the aid of tables of certain trigonometric functions. Since observational values could only be measured with limited accuracy, it was sufficient to give approximately the quantities to be calculated. This led gradually to the invention of terminating decimals, which proved much more suitable for practical computations than the common fractions. Most probably the conviction grew that the results would be the more exact the more decimal places used, and even that every preassigned accuracy can be achieved by using sufficiently many decimal places. In the last analysis this approach grasps the very essence of the real numbers; indeed, mathematicians no longer shied away from talking of decimal fractions with infinitely many places. If this theory had been developed consistently, the result could have been an exact theory of the real numbers. An interesting example of fundamental significance shows how this notion in a slightly different form appears as early as in Archimedes' work, when he tries to calculate the area of certain parts of the plane with curvilinear boundaries. First of all, in his famous exhaustion method he succeeded in calculating the area bounded by part of a parabola and one of its chords. It turned out that a certain ratio of areas of paramount importance was 1/3. But Archimedes did not succeed in finding a correspondingly simple result for the area of a circle. To solve the problem he would have had to calculate the number:n:. As we now know, he could not succeed, being only in possession of fractions; he had to be satisfied with proving that the number:lt lies between two fractions, namely 31 / 1 and 31 °/71. For this purpose he calculated, by a repeated application of Pythagoras' theorem, the areas of the regular convex polygons with 96 sides inscribed in, and circumscribed to, the circle and gave approximate values for them. Clearly ARCHIMEDES was aware that by taking the number of sides and vertices sufficiently large he could include :n: within ever narrower limits and even calculate it with any prescribed accuracy. But this possibility of determining a number approximately with a prescribed accuracy by means of fractions is a characteristic feature of the real numbers. This feeling of familiarity with the nature of the real numbers became firmly established in the course of time on diverse occasions, for example, - long before the foundation of the differential and integral calculus - in the composition of logarithmic tables, in Descartes' analytic geometry, where the points of a plane or of space are specified by coordinates, and then to a large degree in the development of the differential and integral calculus, which was started by LEIBNIZ and NEWTON and continued, as if intoxicated by the joy of discovery, by the BERNOULLlS, by EULER and FERMAT, by CAUCHY, GAUSS and others. No one imagined that the foundation of the theory of the real numbers would require a further intensive study. However, questions of foundation played their part in two other branches, geometry and algebra. As already indicated, Euclid's geometry takes as its starting point a system of very simple geometric propositions from which further theorems of geometry can be derived. These simple propositions, called axioms, represented an extract of the geometric knowledge of the time and were intuitively so Flear that no one felt the need to prove them. An exception was the parallel axiom (or postulate). This states that to a given line and a given point not on the line there is one and only one line passing through the given point without intersecting the given line. Was it perhaps possible to remove this statement from the system of axioms by deriving it from the remaining axioms? - For 2000 years mathematicians wrestled in vain with the problem, until GAUSS in Germany, LOBACHEVSKII in Russia, and B6LYAI in Hungary succeeded in showing that the parallel axiom is independent of the other axioms. The significance of this result only becomes clear in connection with other developments. In algebra the formula for the solution of quadratic equations can lead to the expression V-I, which at first sight is meaningless. But as long as one calculates with it just as with ordinary roots like V2, V3 or even V:n:, the results invariably makes sense. This strengthened the belief in the right of citizenship of this formation V-I, for which the notation i had meanwhile been accepted. Nearly 300 years elapsed before GAUSS and others showed that what one had done until then can be interpreted in a completely sensible manner as an extension of the domain of real numbers in which there exists a new number whose square is equal to -1. Even GAUSS was so thoroughly familiar with real numbers that he had no scruples in using them

13 without justification. Only when certain difficulties emerged in the process of clarifying the concept of limit at the hands of CAUCHY and other mathematicians of the time, did the real numbers become an object of serious thought. It was recognized that a theory of the real numbers can be founded, in fact in different ways, on a reduction to fractions. The latter, in tum, could be reduced to the natural numbers, and again it appeared that in the domain of natural numbers all their properties could be united in a few perfectly obvious fundamental facts, the Peano axioms. With this reduction to the natural numbers a basis was given for the theory of the real and complex numbers, and also for the whole of real and complex analysis and beyond, even for geometry; for in analytic geometry it is shown how to master the basic objects of geometry, above all the points, by means of their coordinates, which are real numbers. In this context another development should be mentioned, which started rather tentatively about ISO years ago. It was common knowledge that some rules for the multiplication of numbers and some for the addition show a strong formal similarity. Similarly, quite simple formal laws were observed in other mathematical operations, for example, in carrying out several motions in succession. But only very slowly did mathematicians proceed to the next logical step of extracting the common basic properties and of deriving from them new and ever deeper properties by purely logical processes. This field developed gradually to the present-day theory of groups, and again one sees, just as in Euclidean geometry, the emergence of an axiom system with all the subsequent developments. Nowadays large parts of mathematics, above all algebra, but to an ever increasing extent analysis and geometry, are built up axiomatically. The procedure is roughly as follows: given is a collection, usually called a set, of mathematical objects, the elements of the set, together with some system of axioms that describes the basic properties of these objects. Now the following tasks arise: first of all, to draw the most far-reaching conclusions from the axioms, in other words, to carry the theory of such a structure as far as possible; next, to gain a survey of all specific ways of realizing the axiom system in question. It can happen that essentially there is only one possibility of realization, or several, or perhaps even infinitely many; it is also possible that no such realization can be found, for example, when the given axioms contradict each other. If there are several models, that is, ways of realizing the axioms, then one searches for characteristic features by which the various possibilities can effectively be distinguished in finitely many steps. For some structures these tasks have been solved completely, for others we are still far away from a solution. This indicates, incidentally, how closely interwoven axiomatics and mathematical logic are. Even more imperative became the demand for an efficient mathematical logic when at the turn of the century contradictions arose in one of the new structural theories, the theory of sets. Set theory is the simplest structural theory, inasmuch as it is concerned with completely arbitrary collections whose elements are not subject to any axioms, such as points, numbers, motions, functions, figures, but equally well men, stars, chairs or what have you. Since no structural assumptions are made, two such sets are to be regarded as equivalent or equipotent if they have equally many elements. In the case of finite sets the meaning ofthis is immediately clear to everyone; but it was a magnificent achievement to define even for infinite sets something like the number of its elements, the so-called power or cardinality. True, this fails to have some of the properties with which we are familiar when the number of elements of a finite set is involved. For example, in this sense there are just as many natural numbers as there are fractions, but not as many fractions as real numbers, and the set of points on a line has the same cardinality as that of the points in the plane. All these are things which in spite of their apparent lack of intuitiveness are entirely unobjectionable from the point of view of total mathematical rigour. Contradictions appeared, however, in the unrestrained formation of sets; for example, the concept of 'set of all sets' is contradictory in itself. Nevertheless, this was not a crisis in mathematics, as the phenomenon was sometimes called; on the contrary, mathematicians took occasion to reflect more thoroughly on what is involved in defining mathematical concepts. Indeed, a systematic mathematical logic was developed, and today one knows precisely how to avoid such contradictions. One might think that this utmost abstraction, in the form ofaxiomatization of the very general structural theories and of mathematical logic, could lead further and further away from down-toearth applied mathematics. This is by no means the case; it was no accident that l.EIBNIZ, who apart from his immediate creative mathematical work occupied himself with some fundamental questions of logic, has already constructed a workable calculating machine. The appearance of factory-made calculating machines, operated by hand or by a motor, did not give rise to important discussions of principles. But this state of affairs changed radically with the creation of electronic computing machines, by which the speed of calculation was increased drastically. True, these machines work on a simple black-white principle, because in each of their components current does or does not flow. Nevertheless, they can cope with calculations that otherwise would be practically impossible: they perform huge numbers of the simplest operations with an unimaginable speed and so can go through a complicated and protracted program in an acceptable time. Naturally, the duration of such a calculation depends on the skill that goes into the making of the program.

14 After some preliminary work that had been done before the invention of electronic computing machines it soon turned out that in programming certain regularities are observed, which also play a role in mathematical logic, for example, in the theory of algorithms. This once more demonstrated the practical advantages of certain purely mathematical investigations which had been carried out merely for theoretical needs - a truly classical example of the close natural relationship between pure and applied mathematics, in this case computing techniques. In this context it seems appropriate to draw attention to the difference between the theoretical and the practical solubility of a mathematical problem. Quite frequently in mathematics it is not individual, numerically given, problems that are discussed, but general problems depending on certain data, whose numerical values can be chosen in many, as a rule infinitely many, ways. A simple example: to determine the area of a triangle depending on the lengths of its three sides. There is a formula for this area that is valid for all triangles, although there are infinitely many possibilities for the length of each side. Such a problem is regarded as solved when a formula, an algorithm, can be given by means of which the solution can be calculated in each individual case. Here one postulates that the formula or the procedure leads to the numerical result in finitely many steps. When this is the case, a pure mathematician considers the problem as solved. Nevertheless, in practice the problem can still be insoluble if the number of necessary steps is finite, but for reasons of time or economy is too large. This can lead to new and interesting problems of pure mathematics: to find more effective procedures - unless one is satisfied with approximate solutions or one builds faster computers. An enormous step forward in this respect was the invention of the electronic computing machines. It had the consequence that new branches arose, above all in applied mathematics, branches which had not been developed previously, because it was clear from the outset that their main problems could not possibly be attacked and solved within a practically acceptable period. Two examples of problems soluble in principle are the games of 'nine men's morris' and chess. They are soluble in principle, because by the rules there are only finitely many possible games. Nine men's morris is also solved in practice, in that one can give to the first player exact instructions how to react to all possible moves of the opponent so as to win in every case. The same question, whether in chess the white player can always win, is still unsolved in spite of the finiteness of the problem; even if all electronic computers at present available in the whole world were used solely to solve the chess problem, a solution could not be reached: this would require computers working unimaginably faster than the present ones. The development of mathematics, which has been roughly sketched here, led from the simplest fundamental concepts of number, operation, figure, and measure to its present-day thoroughly axiomatized form of a wealth of highly abstract structures and to the modern computing automata whose possibilities are far from being exhausted. A comparison of this development with the table of contents of this book indicates many direct and indirect relationships. Thus, the material of the first part' Elementary mathematics' agrees to a large extent with mathematics as it was developed from antiquity through the Middle Ages and before the foundation of the differential and integral calculus. Only here arithmetic, the theory of numbers, and geometry are not set forth side by side, but one after the other. We begin with the natural numbers, together with the rules for the elementary operations, just as they present themselves as perfectly obvious to a naive person. But the axiomatic build-up follows immediately, starting from the natural numbers and leading up to the complex numbers. Even for these simple concepts a notation is used that was unknown to the Greeks and whose absence was one reason for an extremely cumbersome and unwieldy presentation: the use of letters for numbers. Today it is taken for granted in schools. Here the notation is admirably suited to the basic mathematical concepts, but it is so easy to handle that sometimes there is the danger of thoughtless and mechanical manipulation of letters. This suggestive effect must be strongly opposed, especially in schools: the primary thing is the mathematical idea, and the computational working details are secondary - not the other way round. On this theme GAUSS wrote to SCHUMACHER in a letter of 1 September 1850: 'It is a characteristic of modern mathematics ... that in our language of symbols and names we possess a lever by which the most complicated arguments are reduced to a certain mechanism ... How often is this lever handled just mechanically, although in most cases the authority to do so implies certain tacit assumptions. I postulate that in every application of the calculus, in every use of concepts, one should always remain conscious of the original conditions and should never regard results produced by the mechanism as mathematical property beyond the clearly permitted limits.' Many tasks require unknown quantities to be determined from given quantities. As a rule, the use of letters enables us to state such tasks simply and lucidly. It then happens frequently that problems which at first sight appear totally distinct have one and the same form in the resulting equations or systems of equations. This points again to the parallelism between the mathematical formulation of problems and the abstraction that consists in disregarding the meaning of the given and the required quantities and leaving only the mathematical nucleus.

15 A characteristic feature of modern mathematics is functional thinking. This means that one is concerned with functional relationships, such as the dependence of certain quantities on certain others, for example, the area or the angles of a triangle on the lengths of its sides. We shall become acquainted with other examples of this kind of thinking in the analysis of the notion of a function. Elementary geometry deals with points, segments, angles, straight lines, triangles, quadrangles, circles, tetrahedra etc. in a plane or in space. An essential role is played here by the concept of number as developed previously, owing to the need for measuring the objects. Naturally, this must not lead to a neglect of pure geometrical thinking, especially in the solution of problems. One tries to solve geometric problems by purely geometric means, that is, by constructive drawings. How to treat problems in space by drawings in the plane is the topic of descriptive geometry. The most intimate fusion between geometry and calculation occurs in analytic geometry: by means of the concepts of coordinates geometric problems can be transformed into numerical problems: in this way geometry becomes accessible to the far-reaching methods of analysis. The rudiments of analysis itself are treated in the second main part 'Steps towards higher mathematics '. Although the concept of limit is already used in elementary mathematics in an intuitive fashion, higher mathematics begins just with a rigourous theory of limits. This in turn is the basis, on the one hand, for the theory of infinite series of numbers and functions, on the other hand. for the notion of continuity of functions as well as for the differential and integral calculus, whose significance is fundamental not only for the entire framework of mathematics, but also for the applications in physics. technology etc. Many problems of geometry and physics present themselves in the form of differential equations. that is. in relations between a function and its derivatives. The theory. which has grown by now to a very large volume. can only be sketched here in its 'simplest parts. An attractive branch is differential geometry, an application of the differential and integral calculus to the theory of curves in a plane and in space and to surfaces in space. As we remarked above. the theoretical solution of a problem is frequently far removed from an immediate application of specific cases. because the necessary numerical calculations become too extensive. It is the task of graphical representations and of numerical methods to transform theoretical solutions into directly applicable ones. Probability theory and statistics also play an important part in applications. In the last main part •Brief reports on selected topics' an attempt is made to give an insight into a number of research fields of contemporary mathematics. For the reasons stated at the beginning, a more detailed account of the individual problems is impossible, and domains that at present are still in a nascent stage or in the process of deep reorganization could not be included. The reader who wishes to acquaint himself more thoroughly with one branch or another would do well to refer to the specialized literature - and this applies equally well to the first two main parts.

Hans Reichardt

Authors and translators

The authors of the 'Kleine EnzykIopadie der Mathematik' are: Dr. S. Oberlander G. Berthold Prof. M. Peschel Prof. O. Beyer Dr. G. Pietzsch Prof. L. Bittner Dr. B. Renschuch Prof. H. Boseck Prof. H. Sachs Dr. H. G. Bothe Prof. H. Salie Dr. G. Czichowski H. Schlosser J. Dahnn Dr. E. SchrOder Dr. C. Frischmuth Dr. L. Stammler Dr. D. Gohde A. Steger W. Gohler Prof. R. Sulanke Prof. L. Gorke Prof. H. Thiele Dr. M. Hellwich Dr. H. Herre Dr. H. Thiele Prof. M. Herrmann Prof. W. Tutschke H. Kastner Dr H. Vahle G. Lisske Dr. L. Wagner Dr. G. Lorenz Prof. W. Walsch Dr. G. Maess Dr. V. Wiinsch Dr. W. D. Miiller Dr. G. Wussing Dr. F. Neigenfind Prof. H. Wussing Prof. F. Nozicka The present English version of the 'Kleine EnzykIopadie der Mathematik' was prepared under the editorship of Professor K. A. Hirsch and with the collaboration of Dr. O. Pretzel Dr. E. J. F. Primrose Professor G. E. H. Reuter Dr. A. Stefan Dr. A. M. Tropper Dr. A. Walker

I. Elementary Mathematics

1. Fundamental operations on rational numbers 1.1.

1.2. 1.3.

The natural numbers N . . . . . . . . . . . Numbers and digits . . . . . . . . . . . . . . . Calculations with natural numbers N Elementary Number theory . . . . . . . . The integers Z . . . . ..... .. ... .. .. Foundations . . . . . . . . . . . . . . . . . . . .. Calculations with the integers Z . . .. The rational numbersQ .. . ...... . Foundations. . . . . . . . . . . . . . . . . . . . .

17 17 20

Calculations with common or vulgar fractions. . . .... . . . . . . . . . . . . . . . . . Decimal fractions. . . . . . • . . . . . . . . . Computations with decimal fractions

31 33


Proportionality and proportions . . .



Working with numerical variables. Working with algebraic sums. . . . .. Fractions with variables. . . . . . . . . ..

40 41


27 27 28

30 30



1.1. The natural numbers N Numbers and digits

What are natural numbers? Two kinds of activity made our ancestors face the necessity of occupying themselves with numbers; this led to the development of cardinal and ordinal numbers. Cardinal numbers. Man had to compare various sets of things, for example, flints, dogs, hunting companions, in order to ascertain which set contains more elements (constituents, members). Today one does this, as a rule, by counting and comparing the quantities so obtained; this presumes an ability to count, that is, a knowledge of the numbers. But there is an easier way: if one wishes to find out, for example, whether men and horses are present in equal numbers, one simply places a rider on every horse. III other words: one sets up a matching, a correspondence, between men and horses. This matching may tally - then there are just as many horses as there are men, and one says: the sets are equipotent, - or some of one kind are left over; then there are more of this kind (Fig.). In laying a table one arranges a correspondence among sets of cups, saucers, spoons, etc. All sets between which such a matching of pairs can be established therefore have the corresponding number as a common property (Fig.). This is the way in which even today our children gain their knowledge of the cardinal numbers.


Men and horses- without matching

1.1-2 Men and horses- with matching. One man is left over


1. Fundamental operations on rational numbers 1.1-3

Common number: three I, 2, 3, ...

Cardinal numbers are counting numbers

Ordinal numbers are place numbers

l ist, 2nd, 3rd, ...

Abstraction has not progressed this far in all stages of civilization. There are primitive tribes who use distinct numerals when they refer to distinct objects. Two women is then something other than two arrows; here the abstraction of number from the other properties of the sets has not yet been achieved. Ordinal numbers. The second need consisted in creating order within one and the same set. For example, it had to be laid down according to some point of view - say, the height, the age or the bravery of the rider - who would ride first, second, ... at the hunt (Fig.). Something quite similar occurs when one counts through the elements of set; only, the order so obtained is, as a rule, without significance. In this way, there arise the ordinal numbers.


Natural numbers N

10, I. 2. 3, .. ·1

1.1-4 Set of four hunters. unordered. and ordered by height

Cardinal and ordinal numbers have developed in close interconnection and form the two aspects of the natural numbers; frequently the zero (or null) is, by convention, reckoned to belong to them. Numerals aDd number symbols. For the purpose of oral and written communication and of memorizing cardinal and ordinal numbers number words (numerals) and number symbols are reo quired, the latter particularly for abbreviation and ease of calculation (Fig.). The strong similarity between the words for corresponding cardinal and ordinal numbers in all languages or writings is a sign of their close connection. In English most ordinal numbers have the ending ·th (four-the fourth; a hundred-the hundredth). in writing a stop is added (for example, NEWTON was born on 25. 12. 1642). In the United States the month is placed before the day : 12/25, 1642. Because of the great similarity, in what follows it is sufficient to confine our attention to cardinal numbers; corresponding arguments apply to ordinal numbers. basic symbofs:



V aUXiliary symbols. 5

number word: nine number symbol: IHt 1111 or IX or 9 U·S Three symbols for the number word •nine'


example: 1.1-6 Tally sticks



100 L



1000 0


MDCCLXVlll 1768

Roman number symbols

. Representations 01 numbers. !he simplest representations of numbers occur in tally sticks (Fig.), Pieces of wood scored across With notches to record the items. Frequently they were split into halves ~f ~hi~h each ~y kept one. T~e method of strokes, by which Robinson CRUSOE counted days, IS stdl m use, particularly for tediOUS countings. But very soon, when the numbers become larger thi~ rep~ntation loses its perspicuity: it can be restored by appropriate groupings. Somethm; qUite smular occurs when new words for numbers are formed or new symbols are invented: it would be most uneconomical to introduce a completely new word and a new symbol for every number. I~te!-d one composes words and symbols for larger numbers from those of smaller ones, and these buiJdmg bricks themselves have arisen by the combination of units or smaller groups. According

1.1. The natural numbers N


to the method of this grouping and the arrangement of the symbols one distinguishes between addition systems and position systems. Addition systems. The best known example for an addition system is the Roman method of writil)g numbers. Of the basic symbols ten each were combined to the next higher group; in between there are auxiliary symbols (Fig.). By the way. the origin of these symbols is not completely clear. Some of them. for example M (mil/e) for 1000. have been in use in this form only since the middle ages. The Romans wrote CI::> for 1000. The essence of an addition system is that all number symbols are formed by juxtaposition of as few of these symbols as possible (in our case seven symbols. see Fig. 1.1-7). A rule prescribes that the symbol for the larger number always stands to the left of that for the smaller number. An exception to this rule is motivated by the endeavour to use as few symbols as possible. The number nine can be represented as VIllI (5 + 4) or IX (10 - 1). The latter writing is preferred. Therefore. if the symbol of a smaller number stands at the left. then the corresponding number has to be subtracted. not added. However. it is not permitted to place several basic symbols or an auxiliary symbol in front: MCMLIX for 1959; CML (not LM) for 950. An addition system has disadvantages: in general. the number symbols are very long and therefore lack in clarity; when the numbers grow (in the present case. beyond 10 000). one has to keep inventing new symbols to avoid representations of excessive length; written calculations in an addition system are exceedingly troublesome. Position systems. Our present-day position system goes back to the Hindu from whom it came to us by way of the Near East (Arabic digits). In this perfection it is a fairly late achievement in the historical development of representations of numbers. In the system ten individuals (Units U) are combined to a new grouP. a Ten T and again ten of these to a Hundred H etc. However. no new symbols are introduced for these groups of higher rank (as in the Roman system). but they are distinguished by their position within the entire numerical symbol. In the Roman symbol XXX for thirty each of the three letters has the same numerical value 10. and since it is an addition system. the total number is obtained by adding the three individual values. In the symbol 444 for fourhundred-and-forty-four the three digits also have the same numerical value four; but within the total symbol they stand in different places and therefore have different positional values; the rightmost position indicates the units: 321 means: 3 H + 2 T + 1 U. CCCXXI means: 100 + 100 + 100 + 10 + 10 + 1. Since the gathering occurs in groups of ten each. one talks of a decimal system (Latin. decem 10) or a decadic positional system (Greek. deka 10). Accordingly. the Roman number system is a decimal addition system. The number ten is called the base of the system. The positional values are the powers of ten. some with their own names such as 1 million for 106 = 1000000. 1 milliard for 109 • 1 billion for 10 12 • 1 trillion for 1018. There follow 1 quadrillion. 1 quintillion. etc.• each time with six more zeros. Formations such as 1 billiard for 1015 are rarely used; in the U.S.A. and U.S.S.R .• 109 is called 1 billion. 1012 a trillion. and 1015 a quadrillion. It is probable. but not certain. that the choice of ten as a base is connected with the number ten of our fingers. In old measuring units (one dozen. one gross) one finds traces of a vanished duodecimal system with the base 12; the French word quatre-vingt for eighty points to a (non-positional) system with base 20. and the word score for a group of 20 objects is still in frequent use. Our time measures (1 h = 60 min. 1 min = 60 s). as well as the division of the full angle into 3600 recall the sexagesimal system (base 60) of the Babylonians. This system already showed clearly some features of a positional system. But the complete development of such a system was hampered by lack of the consistent use of a symbol for empty places. a zero. The introduction of zero is one of the greatest achievements of the Hindu (around 800 A. D.). Not only 10. 12. 20. or 60 are suitable as bases of a positional system. Every natural number b 1 can serve as base. because then every natural number a ha$ exactly one b-adic representation a = aRb' + a'_lb'-l + ... + alb + ao. in which the natural numbers a" i = O..... n satisfy 0';;;; a, b. The a, are called the digits of a. Every positional system requires exactly b distinct digits. The binary system. Of particular technical importance is the binary system. which is also called dyadic or dual system. In it the position values are the powers of the base 2. that is. 1.2.4. 8. 16. 32. 64. 128•... These position values are considerably closer to each other than those of the decimal system; therefore the number symbols become comparatively long. On the other hand. one only needs two digits: 0 and 1. For the binary unit the notation L is in frequent use: 7 = 1 . 4 + 1 ·2+ 1 . 1 = 1 . 22 + 1 . 21 + 1 • 2° = LLL, 9 = 1 . 8 + 0 . 4 + O' 2 + 1 . 1 = 1 . 23 + 0 . 22 + O· 21 + 1 . 2° = LOOL, 22 = 1 . 16 + O· 8 + 1· 4 + 1 . 2 + O· 1 = 1 . 24 + O· 23 + 1· 22 + l' 21 + O· 2° = LOLLO. This binary system is often used in digital computers.


O. In exponentiation the basis al;ld the exponent cannot be interchanged, as a rule; for example, 23 = 8 =F 32 = 9, in fact, ab = /I' holds, of course, for a = b, but when a =F b, only for 24 = 42 = 16. One distinguishes between even and odd powers, according as the exponent is even (divisible by 2) or odd. Thus, 64, e 16, and generally a2 • are even powers, whereas 67 , e13, and generally a 2n - 1 are odd powers.


Examples: Powers occur in many formulae and laws of mathematics, science, and technology; for example, in geometry 41'r3 /3 represents the volume of a sphere of radius r, (s2/4) 1"3 the area of an equilateral triangle of side s; in physics gt 2/2 is the distance-time law of the free fall, and in the calculus of compound interest b . (r" - 1)/(r - 1) is the formula for an annuity.

Of particular importance are the powers of 10. They are used (in rough estimates or slide rule calculations etc.) to obtain an idea of the order of magnitude of a number or to write very large or very small numbers in an abbreviated and perspicuous form. 100 = 10·10 = 102 , 1000 = 10·10·10 = 103 , a million = 106 , etc.; for instance, 1291000 can be written 1.291.106 or 1291 . 103 • Also units of measurement are represented in the power notation, such as m2 (square meter), cm3 (cubic centimeter), m/s2 (meter per second squared) etc. Powers whose base lies between 0 and 1 decrease when the exponent increases: (1/2)2> (1/2? (1/2)4 ... , but increase when the basis is greater than 1: 22 < 23 < 24 ... They grow very rapidly; the following problem is in the oldest arithmetic book, named after AHMEs (1700B. C.):


Each of 7 persons owns 7 cats, every cat eats 7 mice, every mouse eats 7 ears of barley, every ear of barley could yield 7 measures. How many measures is this? Solution: this is 75 or 16807 measures. Sign of powers. Since negative numbers can be multiplied, the basis of a power may be negative; by the standard sign rules one obtains, for example, (_3)4 = (-3)· (-3)· (-3)· (-3) = +81 or (-S)3 = (-S)· t-S)· (-S) = -12S. It is immediately evident that the product of two negative factors is positive, that of three negative, that of four pOSitive, and so on alternately. If the number of minus signs is even, the power has a positive value, if the number is odd, a negative value. The exponent indicates the number of (equal) factors. A power with negative basis has a positive value for an even exponent and a negative value for an odd exponent.

2.1. Calculations with powers and roots


To make the essence of this rule quite clear one chooses the basis (-I). Together with the obvious fact that for a positive basis the power is positive one obtains for every positive integer n: 1 (4 1)" =-1- 1,

(_ 1)2" =-+ 1,

- 1. 1

(_ 1)2"- 1

Multiplication and division of powers. Powers whose basis and exponent are distinct cannot be contracted on multiplication or division, for example, a4 c 3 /x 7 • Powers with equal exponents. If one raises a product to a power, for example (ab)", one obtains n factors a' b, altogether 2n factors, namely n factors a and n factors b alternately. Since the factors may be interchanged (commutative law), the product can be rearranged as a product of n factors a and n factors b.


Examples: I. (2xyz)' = 2'x'y'z' = 32x'y'z'. 2. (3a)3 = 3a' 3a . 30 = 3·3·3' a . a . a = 33 a3 = 27a3 . 3.2 8 . 57 = 2.27 . 57 = 2· (2' 5)7 = 2, 10 7 = 20000000.

(a' b)" -

a" .

b"· 1

product is raised 10 a pofttr braising e try raclor to tilt same PO"ft'

Fint law for POfttrs.

and multlpl ing the pofttn 10 obtained. Com'ersely. PO"tn with the same apooent art mult1,ued

b raising the product or the bases 10 the power ghen b the common exponent.

Similarly, a power (a/b)" whose basis is a fraction is obtained by multiplying n equal factors alb, hence is a fraction whose numerator consists of n factors a und whose denominator of n factors b, that is, an/boo 5 )J 5 5 5 5.5.5 Examples: 1. ( '6 = '6''6.'6 = ~


= 63'

125 216'




5X)3 53x 3 12Sx 3 2. ( To = 2J a J = ~. 174 174 3. 34' = 34.34'"

( 17 \ 4




= 14' 14l = 14' T





A f raction (quotient ) is raised to a powu by raising the numerator (diVidend) and iknomiNllor t ame power and dividing the po';>'ers so oblained, Conl)('rsely. powt!rI divisor ) individually to he with the same e xponent are divided by dit'iding th ir base and raising the quotient so obtained t o the po'M.'t!r gi,,'t!n by the common exponent.

Powers with equal basis. By the definition of the power the multiplication of two powers am and a" with the same basis a means that m factors a are to be combined with another n factors a; one then has m + n factors, that is, the (m + n)th power. Examples: 1. 3'" . 31 = (3' 3 . 3 . 3)' (3 ' 3) = 3·3' 3 . 3 . 3 ' 3 = 34 1 = )6. 2. 56a'b ' 9 a'b' . 14a 1 b 3 = 23 . 7· a'b' 2· 71 , a7 b' . 2 . 7 . a1 b3 = 23 1 1.71 1 la' 7 Ib l , 3 = 2' , 74 al4 b9 ,

aM. a"

a ..... '

Second law ror powers: PO"erI .ltb the same basis are multiplied by ralslllltbe bas to the po er by the sum of the aponents. Division. Since the result of every division can be regarded as a fraction in which the dividend is the numerator and the divisor the denominator, one obtains on dividing the power am by the power a" a fraction with m factors a in the numerator and n factors a in the denominator. If n is the smaller exponent, then after n cancellations the denominator becomes 1, and the numerator g1VetI

has n factors a fewer, that is, only m - n factors, hence the value am-no On the other hand, if m is the smaller exponent, then the numerator becomes 1 after cancellation, and there are n - m factors left in the denominator; one obtains I/a"-m. If the two exponents are equal, then both numerator and denominator become I after cancellation, and the division leads to the value 1 for every basis. I

I 6.. Exampes: l .7:7

= I

1, I I 3 : 11 '



'H ' II ' II -';:l'l -=--'''i'i':'''''''''.'W '-:-'-:l:-;'I-'"';17" 1





7.7'7'7.7.7 7'7.7,.7 I I 1 1 1 1 J


76- 4

= 71 = 49.



a M - " when m

u'" I = - - when 0"


= JlW'" =


"'jjT =


"ITi'" '

0" '"

u" = I when m a"


> "

n> "'



2. Higher arithmetical operations

Compared with the result for the multiplication of two powers with equal basis the result obtained for division is unsatisfactory : there the product had the sum of the exponents, here one of the differences m - n or n - m occurs as the exponent for the quotient, or even the number I, which at first sight has nothing to do with powers. Since division is the inverse operation of multiplication, one should expect that the result is determined in every case by the difference m - n of the exponent m of the numerator and n of the denominator; this would lead to the third law for powers. Third law for po1l'en: P01l'ers with equal basis are di ided by raising the ba i to the exponent given by the dlfl'erence of the exponents.

According to the so-called principle of permanence, which was formulated in 1867 by one tries to retain the validity of calculating rules, but to extend the concepts of the mathematical objects connected by them. The difference m - n of the exponents, which occurs in the third law for powers, has a meaning in the first instance for m n only. If, in accordance with the principle of permanence, this law is to remain valid also for In = n and for m n, then the exponent 0 or negative exponents occur, which have no meaning under the definition adopted hitherto: •aO means n equal factors a'. Therefore one extends the notion of power by the following two definitions :



Then 0"': a" For one has 1. for 2. for 3. for

b or b a is defined to hold if there exists a natural number c + 0 such that a = b + c. It can be shown that the relation so defined is an irreflexive order (see Chapter 14.). It satisfies the monotonic law of addition and multiplication as well as the Archimedean axiom.

An arbitrary nest ollnterYals (Q " Q;> wIth real end-points cktmnines uniquely a real nlUDber Q with tbe pro~rty Q, ~ Q .;;; Q ; lor all i. The proof is omitted here. The nest of intervals {(a/I, a;b;)} therefore determines a real number, which is denoted by ",P. The arithmetical operations so defined satisfy the same laws as in the domain of rational numbers. The domain of real numbers is on extension of the domain of rO fiof/ol numbers. If contains. in particular, all rOO fS of positlt·(, numbtrs. The construction of the real numbers could only be sketched here. It should be mentioned that in a formal treatment it is necessary to define an equivalence relation for nests of intervals and to form the corresponding classes. This method, carried through rigorously, is a genuine constructive way of obtaining the real numbers. Other methods of defining the real numbers. EUDOXUS (about 408-355 B. C.) can be regarded as a harbinger for the development of a theory of the real numbers. His geometrically orientated ideas were taken up by Karl WEIERSTRASS (1815-1897) and Richard DEDEKIND (1831-1916) and were further developed, utilizing modern arithmetic and analytic methods. The method of nests of intervals goes back to WEIERSTRASS. DEDEKIND introduced the real numbers by cuts in the domain of rational numbers. Georg CANTOR (1845-1918) constructed them by means of Cauchy fundamental sequences. The domains obtained by these methods can be mapped onto one another by order-preserving isomorphisms. Therefore, structurally there is only a single domain of real numbers.



3. Development of the number system 3.6. Continued fractions



Continued fractions of order n. Let bo , blo b 2, ... , b. be integers with bt 0 for k O. The continued fraction of order n with the denominators b l , b 2 , ... , b. and the initial term bo is defined by the following expression, which is also abbreviated as [b o ; b l , b 2 , " ', b.l:

= 3; b 2 = I; hJ = 4 [2 ; 3, 1,4) = 43/ 19 1/ 4 )] .

Example : n = 3; ho = 2, hi

2 + 1/ [3

+ 1/(1 +


Approximating fractions. Let IX be a continued fraction of order n. By an approximating fraction for IX of order k (k n) one understands the continued fraction breaking off at the kth denominator.

3.7. The complex numbers C In the domain of real numbers the arithmetical operations of the first and second kind can be carried out without restriction. This is not the case for the operations of the third kind; for example, n


the power al/n = ya does not exist when a is negative and II is even: there is no real number V-4. But square roots of negative real numbers are needed, for example, in the solution of a cubic equation by Cardano's formula (see Chapter 4.), in fact, just in the so-called casus irreducibilis, when there are three distinct real solutions. In order to remove this restriction the number system is extended once more. Construction of the new numbers. One considers ordered pairs (a, b) of arbitrary real numbers and b. This time the equivalence relation is the ordinary identity, that is, the pair (a, b) is called equivalent to (a', b') if and only if a = a' and b = b'; every equivalence class consists of a single number pair. Such a pair (a, b) is called a camp/ex lIumber. The arithmetical operations of the first and second kind are defined by:








a2, b l





(ala2 -

bl b 2 , a l b 2


b l a 2).


3. Development of the number system

It is easy to verify that subtraction and division are the inverse operations to addition and multiplication. The commutative, associative, and distributive laws hold for addition and multiplication. For example, the distributive law is proved as follows: let z, = (a" b,), i = 1,2, 3, be three complex z3l = (aI' b1)(a2 a3, b2 b3) numbers. Then: ZtlZ2 = (a1a2 a1 a3 - b1b2 - b1b3 , a1 b2 a1b3 b1a2 b1a3) ' On the other hand, ZlZ2 ZlZ3 = (a1a2 - b1b2 , a1b2 b1a2) (a1a3 - b 1b 3 , a1b3 b1a3) = (a1a2 - b1b2 a1 a3 - b1b3, a1b2 b1a2 a1b3 b1a3)' The two expressions are identical. Furthermore, all the laws for these operations that hold in the domain of real numbers are also satisfied here, except those in which an order relation 'greater than' occurs. There are many ways of introducing a total order in the domain C of complex numbers, for example, first by absolute value and for equal absolute value by argument; but it can be proved that no relationship of total order in C is compatible with addition and multiplication.


+ +







+ +





Complex and real numbers. The domain of complex numbers contains as part that of the numbers

(a, 0), which is isomorphic to the domain of real numbers with respect to the permitted operations: (a, 0) (a', 0) = (a a', 0) and (a, 0)' (a', 0) = (aa', 0). One can therefore treat such numbers by writing simply a instead of (a, 0). The numbers (0, b) are called purely imaginary. In particular, the complex number (0, 1) = i is called the imaginary unit. Brackets for the arithmetical operations can now be omitted, because errors need not be feared. One has (0, b) = (b,O) . (0, 1) = bi and (a, b) = (a, 0) (0, b) = a bi.






Next, i· i = (0, 1)· (0,1) = (-1,0) =-1.


Imaginary unil i






Ev ry comple.r nllmbu can be represented as tire sum of a real and a pllrely imaginary number: the imaginary part of:.: a alld b being real numbers .

= a + bi. Hue a is called lire real part and b

Graphical representation of the complex numbers. In the plane one draws a Cartesian rectangular system of axes and marks on the x-axis the real numbers in the usual way, on the y-axis the imaginary numbers with i as unit. To the complex number Z = a + ib one assigns the point z with the coordinates (a, b) or the vector 4 leading from the origin to this point. These correspondences are one to one. To the sum Zl + Z2 there corresponds by vector addition (according to the parallelogram rule) the vector 41 + 42 (Fig.). To give a geometric interpretation also for the product one represents z = a + bi in terms of the length r of 4 and the angle 'I' which this vector forms with the positive x-axis; r is called the absolute value or modulus and 'I' the argument or amplitude of 4 (Fig.). It should be observed that 'I' is determined only up to multiples of 2n and is measured anticlockwise.

Z" Z;






+2; +i





+1 +2 +3 +4-

3.7-1 Addition numbers


+, "2


0 = rco rp b = r in rp : = a + bi


Ol -+-




,.f.,.. .... I I

l, \

/ \ ,




b2 ,

r , O real sin rp = blr co rp = aIr, z = r(co rp + j in'll) rl


3.7-2 Modulus and argument of a complex number

Transformation formulae

,, \


-2 i

of complex


3.7-3 Multiplication of complex nUIllbers

The product ZlZ2 = r1(cOS'l'l + i sin 'I'd r2(cos '1'2 + i sin '1'2) can be transformed by means of the addition theorems for sine and cosine into ZlZ2 = r1r2(coS ['1'1 + '1'21 + i sin ['1'1 + '1'2])' This representation leads to the following geometric interpretation (Fig.): the triangle formed by the points 0, Z2, and Zl . Z2 is similar to that formed by the points 0, (+ I), and Zl, because the angle

3.7. The complex numbers C


rpl is common to the two triangles and the sides including it have the same ratio '1'2 : '2 = 'I : 1. Thus, there is a simple geometric construction for the product.

Powers and roots. As usual, z" for a natural number n is defined by ZO = I, zn+ 1 = z" . z. By means of the addition theorems and ~athematical induction one. derives the de Moivre' formula z" = roCco 1Kp + i in I/rp) Important formula of de MOIvre: For n = -lone obtains Z-I = , - I (cos (-rp) + i sin (-rp» = ,-1 (.cos rp - i sin rp) and by mathematical induction z-" = ,-"(cos (nrp) - i sin (nrp».




By one means a complex number w whose nth power is equal to z, that is, a solution of the equation w" = z. Let w = e(costp + i sin '1'). Then from w" = z = ,(cos rp + i sin rp) it follows by de Moivre's formula that :


e= 'I' = rpl n + k ·2nl n or W = }tr[cos(rpl n + k· min) + i sin(rpln + k . m i n»). =l= 0, then n distinct values arise for k = 0, 1, 2, ... , n - 1. In the domain of complex numbers is not restricted to a single value, but is many-valued. How this many-valued ness the symbol can be mastered is shown in the theor.y of Riemann surfaces (see Chapter 23.). In particular, when z is a positive real number, then the uniquely determined positive real nth root is called the principal value. The extraction of roots can be performed without restriction in the domain of complex numbers. Among the n values of ~z is the number rl/"(cos (rpl n) + i sin (rpln» .

e" = "




• I), one set : Example: To obtain all alu of V(z = I [co (180° + k . 360°) + i sin (I 0° + k . 360°») W = I [co (180°/4 + k· 90 + i in (180°/4 + k· 90°)) k = 0 gives Wo = co 45° + i sin 45°, k = I giv WI = co 135 i in 1350 , k = 2 gives WI = co 225° + i in 225°, k = 3 giv Wl = co 315° i in 315°. For k = 4 one ha '1'4 = 405° = 360° + 45°, hence 1 0 the equation x 2 + q = 0 with q = R cannot have a solution with x E R, because in this case the expression on the left-hand side always satisfies x 2 + q > o. For q < 0, hence (-q) > 0, the expression x 2 + q can be written by means of the binomial formula a 2 - b 2 = (a - b) (a + b) as a product of two linear expressions in x : x2 + q = x 2 - (V_q)2 = (x - V-q)(x + V-q). Consequently, the equation (x - V-q) (x + V-q) = 0 is equivalent to the given one. Since the product El . E2 of two expressions El and E2 is zero if and only if El = 0 or E2 = 0, it follows from the equivalent equation that x - V-q = 0 or x + V-q = O. So the solution of the pure quadratic equation is reduced to that of two linear equations. From the first equation one obtains Xl = V-q, and from the second X 2 = -V-q. Hence the given equation has two solutions Xl and x 2; this is expressed in the combined solution formula Xl.2 = ± V-q. The solution set Sis the union of the solution sets of the two linear equations, that is, S = {V-q, -V- q}. Check. I. (±V_q)2 + q = 0, 2. ± V-q is real provided that q o. -q+q= 0, 0=0 true But if one chooses for the domain of variability the set C of complex numbers, then for q 0 there exist two imaginary solutions which differ by sign only and which can be obtained formally in the same way, by splitting the expression x 2 + q into (x + V-q) (x - V-q).


Pure quadratic equation



qE R

solution formula Xl,l

= .... 1 -

xE R

q .;; 0; olution real

q > 0; no real solution



q > 0; solutions imaginary


4.3. Quadratic equations Examples:

144 = 0 xll =± I - I44 2}. For x E R there i no solution, because X I .2 V- I44¢ R; hereS = 0. Check : I. ( 2)2 - 4 = 0 For x e C one has XI . 2 = ± 12i and S = {- 12i, + 12i}. 4 - 4 = 0 true Check : I. (± 12i)2 + 144 = 0 2. + 2 e R true - 144 144 = 0 true - 2 e R true 2. ± 12i e C true 10. Solution of • ~xed . quadratJe _~uado~ wJ~t ~~Iute t~rm. Taking x before a bracket transforms the equation x~ + px = U mto the equivalent equation x(x + p) = O. From this it follows that x = 0 or x + p = O. The first of these two linear equations has the only solution Xl = 0, the second X2 = -po Hence the mixed quadratic equation without absolute term always has two real solutions of which one is zero; the solution set is S = {O, -p} . Check : 0 2 + p . 0 = 0 true for all pER; (_p)2 + p(_p) = 0 true for all peR .

I. x 2


X l.2

2. xl

4 = 0; x e R 14 2; S = {- 2,

Mixed Quadratic cquation ithout ab olute term £'(ample:


2.'( = 0 x(7x - 2) = 0 x(x - 2j7) = 0 7X2 -


+ p = 0; p '" 0 = {0,2j7}

xe R pe R

S = (O, - p )

Check: for XI : 7 . 0 - 2 . 0 true and 0e R for Xl: 7· (2/1)2. - 2 ' 2/7 = 0 4/7 - 4/7 = 0 true and 2/ 7 e R

true true

IV. Solution of a mixed quadratic equation X2 + px + q = O. The idea of the solution is to make the expression x 2 + px into a perfect square by adding a suitable term, and so to reduce the equation to a pure quadratic one ; here the quadratic supplement is the square (P/2)2 of half the coefficient of the linear term px in the normal form. To make an equivalent transformation of the given equation, one has to add (p/2)2 - (p/2)2 . For example, for the equation x 2 + 2x - 5 = 0 one has p = 2 and (p/2)2 = 1. By addition of 1 - 1 this equation goes over into x 2 + 2x + 1 - 5 - 1 = 0 or (x + 1)2 - 6 = 0, that is, into a pure quadratic equation in (x + 1), and from its solutions those of the given equation can be obtained. From (x + 1).,2 = ±)l6 it follows that XI , 2 = -1 ± )16. To achieve that in the following solution method the pure quadratic equation is always soluble, for a while the domain of variability is taken to be the set C of complex numbers. Solution method: x 2 + px + q = 0, I + (p/2)2 - (P/2)2 quadratic supplement: x 2 + px + (P/2)2 - (P/2)2 + q = 0, pure quadratic equation: (x + (p/2»2 - [(P/2)2 - q) = 0 its solution: (x + P/2)"2 = ± )I(P/2)2 - q), solution formula: Xl. 2 = -p/2 ± )I[(p/2)2 - q). Check : 1. [-pj2 =t= )I[(Pj2)2 - q]2 + p{ -pj2 ± )I(pj2)2 - q]} + q = 0, (pj2)2 ± P )I[(pj2)2 - q) + (pj2)2 - q - p2j2 ± p )I[(p/2)2 - q) + q = 0 , p2j4 + p2j4 - q - p2j2 + q = 0, 0=0 true 2. -pj2 ± )I[(Pj2)2 - q) e C true Mi ed quadratic equation

Di riminant

x 2 ..,. px p,q e R


0 = (p/ 2)2 - q

xe C

olution formula = - p/2 ± H(p/2)2 - q)


xe R


0 > 0

.'(1,2 = - p j 2


0 = 0


III 0 < 0

= Xl =

± VD - p/2

no rea l solution

The solution formula also contains the solutions for the special cases, as one can verify by substituting p = 0 or q = 0 or both. It is applicable whenever the equation is given in its normal form . Discriminant. Evidently the nature of the solutions of the quadratic equation is determined by the radicand 0 = (Pj2)2 - q of the root in the solution formula. It is called discriminant. If p and q


4. Algebraic equations

are real parameters and if one returns to the set R of real numbers as domain of variability, then three cases are to be distinguished : I with two distinct solutions, II with two equal solutions and III with no real solution. If one chooses as domain of variability the set C of complex numbers, then two conjugate complex O. Choosing as domain of variability a subset of the set of real solutions occur in the case D numbers, the solution set may be different.

4.3-3 The Brinell hardness test

4.3-4 Section of a hollow sphere

fereomefrie problem. A hollow teel phere has the rna M = 160.72 lb. The all i Ii' = 2.36 in. (Fig.). hat ' it inner radiu r and outer radius R if the Ihicknes of il den ity i Il = 0.2 pound per cubic in h? If the inner radius r h the length X inches then the outer radiu is R = (x + \ ' greater than', ;;;. ' greater than or equal to ', < 'less than ', ~ 'less than or equal to', or + 'unequal to', then E l , E1 ~ E l , or E1 + E l ; for there arises one of the inequalities E, > E l , E1 ;;;. E l , E1 example, 3x 5, a l ;;;. 9, 2 ~ 8, x + y> 6, 1/2 + 1/3 are inequalities. The only inequalities to be treated in what follows are of the forms E1 > El and E1 < E l . Just as for equations, so one distinguishes among inequalities between those without variables, which are propositions on inequality that can be true or false, and those that are predicates on inequality; for example, 2 8 and 1/2> 1/3 are propositions, while a l 9 and x + y > 6 are predicates. Solution set and solution of an inequality. Every number from the domain of definjtion which on substitution for the variable makes an inequality with one variable into a true proposition is caUed a solution of the inequality. Here the domain of definition of an inequality is defined by analogy to that of an equation. If the inequality contains two, three, ... , n variables, then a solution is an

0 and y > -2 are equivalent over N, but not over Z . Transformations carrying an inequality into an equivalent one are called equivalent transformations. They are based on the fundamental laws of arithmetic, especially on the monotony properties of real numbers.





1+ 2a - 25 / ·(- 1)



r'I': 0; x E It 4.7-3 representation of the solution set of the inequality x >' < 4 for x E N, E N and for x E It . >' E It

A product is positive if and only if both factors have the same

ign. This leads to two cases :

Fir t case: Second case: x - 2 0 and x 2 < 0 x - 2 0 and x + 2 0 x 2 and x x > 2 and x >-2



Example 4:



< 0; x e R.


(x - 2) (x

+ 2) < O.

A product of two factors is negative if and only

if the factors have opposite ign. This leads to two cases:

First case : Second case : x - 2 0 and x 2> 0 x - 2 0 and x + 2 < 0 x 2 and x x > 2 and x < - 2 >- 2 - 2

Example 5: (x + 2)/(x - I) 4; x e R . The domain of definition of the inequality is the set of all real numbers x ", I. In fractional inequalities one has to make case distinctions :


4. Algebraic equatiolW

First case: (x 2)!(x -

+ +2>



x 4(x - I) x + 2 > 4x - 4 6 3x x < 2

and and and and


Second case : (x + 2)!(x - I)


and x - I x> 1

x x 6 x

x> 1 x > 1 x > 1

+ 2 < 4(x -

+ 2 < 4x < 3x >2


I) 4

and x and and and and

Jlll bPI ;;;;; U Plla -

;;;;; UPI 6 . + I I z1/(lbIIP 6 I>. Since ! PI> 6 Z is Ibl < 6 Z + IPI. hence lalb - ! PIso;;: UPI t . + I I tzl IlIPl(lPI ..j.. 62»)

0, b > 0 and n = I, 2, 3. .. . one alway has by the binomial b)". theorem a O + b· ~ (a S. Bernoulll's inequality: (I + a)" I + na for natural numbers n ;;;. 2 and real a • 0 and a > - I. 6. For real numbers a ~ O. b ;;;' 0 one always has ab ~ [(a b)/ 2)Z or \/(ab) ~ (a + b)/2; in






general. for n E N and real numbers a. ;;;. O• .. .• a o ;;;. o. V(a.a2 ... a o) ;;;;; (a. az + ... in words : the geometric mean i alway less than or equal to the arithmetic mean. 7. Between the arithmetic mean A

= (a. + a2 + ... + a.)/n. the geometric mean G

+ a.)/n;

=t( ... a o) n and the harmonic mean H = I/a. I/ a 2 -J- Ila. the following relation holds : A ;;;. G ;;a H. where the a. are non-negative real numbers and n is a natural number. 8. Cauchy-Schwarz inequality: for all real numbers a •• az ....• a•• b •• b z • ... . b. one has (a.b. a1 b 2 a.b.)l ;;;;; (at a1 a~) (bI + bl + b~) .


+ ... +


+ ...


+ .. +

+ ...

5.1. Basic concepts


S. Functions 5.1.


Basic concepts .................• Concept of a function ............ Representation of functions. . . . . . .. Special types of function . . . . . . . . .. Inverse of a function ............ . Polynomial and rational functions The concept of a rational function .. Linear functions .......... .. ..... Quadratic functions .............. Cubic functions . . . . . . . . . . . . . . . . . . Power functions with positive exponents . ................. . .. ... .. Polynomial functions ............. Factorization of polynomials . ..... Zeros ........ • .....• . ..... . .... The behaviour ofpolynomial functions at infinity ..... .. .. .. ..... .... .. Power functions with negative exponents ...... ............ .. . .. .. . General form of rational functions ..

107 107

Zeros and poles of rational functions 126 The behaviour of rational functions at infinity . . . . . . . . . . . . . . . . . . . . . . . .. 127 Decomposition into partial fractions 128


111 113 115 115 115 116 118



120 120 121 124 125



Non-rational functions . . . . . . . . . .. Root functions .................. Exponential functions ... . .. . .... • Logarithmic functions .......... . . Trigonometric and circular functions Hyperbolic functions . ........... . The inverse functions of the hyperbolic functions .... . ........ . .... Functions with more than one independent variable ......... . ...... General definition . . . . . . . . . . • . . . .. Real functions with two indepedent variables . . . . . . . . . . . . . . . . . . . . .. Real functions with n independent variables .......................

130 130 131 133 133 134 134 136 136 136 138

5.1. Basic concepts Concept of a function

In accordance with a definition, which EULER had already given in 1749, a function is often explained as a variable quantity that is dependent upon another variable quantity. For many purposes such a definition of the concept of a function suffices. But in the course of the further development of mathematics it turned out to be necessary and useful to give a more general and abstract content to the concept of a function . The essence of the concept is not the dependence of quantities, by which one usually understands numbers that can be compared in a 'less than or greater than' relationship, but the fact of the correspondence itself, on the basis of which certain objects are regarded as being assigned to certain other objects. The concept of a function is reduced to settheoretical definitions. Correspondences. Every metal bar alters its length when heated. Suppose, for example, that a copper bar has a length of 10 = 200 units u of length at O°C, say centimetres or inches, then its length I at a temperature tOC is given by 1= 200(1 + 0.000016t). By this formula each value of t between O°C and 100°C is made to correspond to a certain length I between 200u and 200.32u. Similarly, to each quantity of a merchandise there corresponds a certain sum of money as its selling price, and to each page number in this book, a number stating how many letters occur on the page concerned. Correspondences exist not only between numbers, but more generally between elements a in a set A and elements b in a set B; for example, each seat for a performance in a theatre corresponds to an entrance ticket or to a particular visitor. Thus, the domain of definition r(Jnge correspondence is determined by a relation F defined on A v B (see Chapter 14.) with domain of definition D(F) !;;;; A and range R(F)!;;;; B. If with respect to this relation F one and only one element b of its range R(F) corresponds to each element a of its domain D(F), then the relation is said to be single-valued and one speaks of a function or mapping from the set A into the set B (Fig.). The element b of the range corresponding to the original element a of the domain is called the image of a. Consequently the function F is a set of ordered pairs (a, b) whose first element belongs to the domain of definition D(F) and whose second element belongs to the range R(F). For a mapping of A into Bone has D(F) = A; that is, every element a E A occurs as 5.1-1 Graph of a function


5. Functions

an original element, and for a mapping of A onto B, in addition, every element bE B occurs as an image. The element y that is assigned to the element x by the function f is often denoted by f(x) and the correspondence is then written x - y = f(x), or more briefly y = f(x). The element x is called the argument and the corresponding element y the function value f(x) at the point x. The domain of definition (or just domain) of the function x -. y = f(x) is denoted by X and the range by Y. Iff is a function from A into B, then clearly X ~ A and Y ~ B. A functionfisa mapping from a set A Inlo a set B, thaI Is, a non-empfy sel of ordered pain

(x, y) e f with x E X ~ A, y e Y S;; B and with the property that 10 each x E X there corresponds




e Y.

Representation or runctions To describe a function one must give its domain of definition and its range and the rule for the correspondence. domoin of definition 1 2 3 I, Graph. In the graph of a function the do5 6 7 main and the range are represented diagrammatically and the correspondence is indicated ronge b. by arrows (Fig.). Only one directed line goes out from each element of the domain, but one 5.1-2 Table of values of a function or more of these lines may lead to anyone element of the range. Table or values. The rule for the correspondence can also be set down in a table of values (Fig.) rather than by means of a graph. The elements of the domain are entered ;n the top line of the table and under each one is the corresponding element of the range. A table of "alues can give only finitely many ordered pairs; it is not sufficient for the complete description of an arbitrary function F. Explanation in words. If the domain and the range of a function are not finite or are so extensive that it is no longer possible to represent the graph or the table of values on a sheet of paper, then it is sufficient to give an exact description of the domain and the range, together with a rule by which for every element of the domain the corresponding element of the range can be found. A function can be defined entirely without the use of mathematical symbols, by means of a sentence in everyday language; for example, a function is defined if to every first division match in the football league there corresponds the quotient of the number of entrance tickets sold and the number of inhabitants of the place where the match is played. This function can give a certain indication of the interest shown by the public in individual games. Many examples can be found of rules of correspondences that are formulated entirely or partly in words.

0 0 0 0 0 !l

Example J.- To each real number x there corresponds either the value 0 or the value 1, according as x is irrational or rational. For example, 2 - 0; (3/4) - 1. Example 2: g(x) = [xl, where x denotes a real number and [xl denotes the greatest inleser that i less than, or equal to. x .

Diagram. A diagram likewise represents a function if one chooses a set of numbers of the horizontal axis as domain of definition and a set of numbers of the vertical axis as range, and assigns to the argument x of the domain precisely that value of y for which the point with the coordinates x, y is a point of the diagram. However, not every arbitrarily drawn curve in a coordinate system can be regarded as the representation of a function. The correspondence given by means of the curve must be single-valued. This is the case if the curve of the diagram is cut by each line parallel to the vertical axis in at most one point. Formula. The most frequently used method of representing a function in mathematics is the formula. In this the elements of the domain and range are now only numbers, or at least mathematical objects for which suitable rules of calculation can be given; for example, (1) y = 7x + 2; (2) y = V(x - 4); (3) y = sin x. If no particular information is given about the domain of definition, one usually regards those numbers as belonging to it to which a definite value can be ascribed by means of the formula. In the cases (1) and (3) these are all real numbers, and in case (2) all real numbers greater than or equal to 4. The range is then given by: (1) -00 y +00; (2) 0,;;;; y +00; (3) -1 ,;;;; y';;;; +1. Restriction of the domain of definition. The domain of definition can, however, be arbitrarily restricted, for example, (1). y = 7x + 2 (for -3 ,;;;; x 5) or (1)·· y = 7x + 2 (for -8 x 0), and so on. The range is then given by (1)· -19';;;; y 37, and (1)·· -54 y 2. Here it is essential that, according to the definition of the concept of a function, (1), (1)· and (1)•• represent


5.1. Basis concepts


first function correspond exactly to m periods of the second function. Consequently the sum function has the period m . (2n/al) = n . (2n/a2)' Example: The periods of the individual functions of the sum function y = sin (2x ) + 2 sin (3x/2) are n and 4n!3 and their ratio is n /(4;r./3) = 3/4. The given function therefore has the period 4;r. (Fig.).

Inverse of a function Invertible functions. The single-valued correspondence determined by a function between the elements of the domain and the elements of the range, conversely also assigns to each element of the range one or more elements of the domain. Functions for which each element of the range occurs only once as the image of an element of the domain have a special significance, because the inverse of the correspondence is also single-valued. To each element r of the range there belongs only one element d of the domain. In this case the range of the given function / can 5.1-8 Graphs of the functions y = sin (2x), y = 2 sin (3x/2) be regarded as the domain of a new and y = sin (2x) + 2 sin (3x/ 2) function qJ. If the given function / determines the correspondence d~r=/(d), then for the new function qJ one has r ~ d = qJ(r). In other words, (r, d) E qJ if and only if (d, r) E f Functions for which in this sense the correspondence between the domain X and the range Y can be inverted are called invertible functions (Fig.). These are one-to-one correspondences of X onto Y. Monotonic functions belong to the class of invertible functions: a monotonic function is always invertible. On the other hand, an invertible function need not necessarily be monotonic; for example, the domain and range may not be ordered sets, so that the concept of monotonicity is not defined. Again, a non-monotonic function can also be invertible, for instance if the domain and the range consist of only finitely many elements. An example of this is the function given by the following table of values:

: I ~ ~ : : : 6 : : ~ 1:

5.1-9 Graph of a non-invertible function (left) and an invertible function (right)

Inverse function. If one regards the range Y of an invertible function / as domain of definition of a new function qJ, whose range is the domain X of f, and if one reverses the singlevalued correspondence between the sets X and Y given by the function f, then one obtains the inverse function qJ of the given function f The inverse function is itself an invertible. By considering d ~ r = /(d) and r ~ d = qJ(r) it is easy to see that the inverse function of the inverse function of a given function / is/itself. Thus, one is justified in calling/and qJ mutually inverse functions. Example i :

Function/ domain a range

2 b

3 c

4 d

5 e

Inverse function qJ of/ domain abc range 2 3





If y = /(x) is the equation of an invertible function, then the same equation naturally also de

scribes the inverse function, only y must then be the independent and x the dependent variable

It is agreed, however, that in a function equation of this form x shall always denote the independen and y the dependent variable and, whenever possible, the explicit form of the function equation

shall be given. One therefore rearranges the equation as follows :


5. Functions

1. In the given function equation y == f(x), y is regarded as the independent and x as the dependent variable. 2. Denoting the independent variable by x and the dependent one by y, x = f(y) is an implicit form of the equation of the inverse function. 3. If this equation can be solved for y, one obtains y = !p(x) as its explicit form. Example 2: From the function equation y = x/2 of a given invertible function one obtains y/2 after interchanging tbe variables. Solving for y gives y = 2x. The function y = x/2 has the inverse function y = 2x with the domain of definition -1 .;;;; x .;;:; 2 with the domain of definition - 1/ 2 ';;:; x .;;:; I and the range - l j2 ';;:; y ';;:; l and the range - 1 ';;:; y ';;:; 2. Example 3: From the given invertible function y = 3x + sin x, interchange of the variables gives the function equation x = 3y + sin y , which cannot be solved explicitly for y . Thus, the inverse function must be given in the implicit form 3y + sin y - x = o. x



Graph of tbe inverse function. Because of the uniqueness of the mapping represented by a function, every line parallel to the y-axis cuts the graph in only one point. If the functionf(x) has an inverse function !p(x) and is therefore one-to-one, then each line parallel to the x-axis also cuts the graph of the function in only one point. This curve represents both the correspondence x ...... y and the correspondence y ...... x. Because of the interchange of the variables in the inverse function each particular number pair (a, b) of the function f becomes a number pair (b, a) of the function tp. The points corresponding to these number pairs (a, b) and (b, a) are mirror images of one another in the angle bisector of the first and third quadrants of the Cartesian coordinate system. Consequently the graph of the inverse function !p(x) is obtained by taking the mirror image in this angle bisector of the graph of the given function f(x) (Fig.).


y -tp(x)


y b

:i. I ·1 1 Graph of y = arcsin x ; princi pal x value y = Arcsin x is _+-----:;>F-_+-__

S.I·10 The funclion curve of the inverse fu nction

drawn in black

ln~erses of functions in particular intenals. In the discussion of monoton ic functions it has already been shown, that non-monotonic functions may be monotonic in certain intervals of the domain of defin ition . In these intervals they are also invertible.

Example J: The function y = x 2 is monotonic and invertible in the interval 0 .;;:; x +00. In this interval its inverse function is y = Vx. Naturally it is also monotonic and invertible in the inte.rval - 00 < x .;;:; o. Here the inverse function is y = - Vx. Example 2: The domain of definition of y = sin x can be split up into intervals in which the given function is monotonic. The inverse function is denoted by arcsin x, but in each case the range must be stated, because otherwise it is not clear in which interval of monotonicity the inverse has been formed. For example, if y = sin x is inverted in the interval 3n/2 .;;:; x .;;:; 5n/2, then the inverse function should be denoted by y = arcsin x (3n /2 .;;;; y .;;:; 5n /2). If the range is not specified, then arcsin x is always understood to mean the principal value, which lies in the interval [- Te/2, + n/2] and is denoted by Arcsin x (Fig.). Example 3: Also for the other trigonometric functions intervals can be chosen in which they are monotonic, so that in them circular functions are defined as their inverses. The function y = cos x, for example, decreases monotonically in the interval 0 ';;;; x .;;;; + n from y = + I to Y = - I and in doing so assumes all values of its range exactly once. Hence in this inte.rval an inverse function exists. It is denoted by y = arccos x . Its domain of definition is - I .;;:; x .;;;; + 1 and its range is n ;;;;' y ~ O. If the function y = cos x is inverted in another interval in which it

0 there belongs a cubical parabola obtained from the standard cubical parabola by stretching for k> 1 or contracting for k < 1. Finally, the graph of the function y = (x - a)3 + b is obtained from the standard cubical parabola by translation parallel to the axes of coordinates, with the new centre of symmetry Z = (a, b).



S.2. Polynomial and rational functions


The general cubic function y = Ax3 + Bx 2 + ex + D always has three zeros of which, under certain conditions between the coefficients, two can be conjugate complex. In the differential calculus it is shown, in addition, that when it has three real zeros, the function has two extrema, one (local) maximum and one (local) minimum. The example shows that the graph of such a function cannot be obtained by simple transformations from the standard cubical parabola y = x 3 (Fig.). Example: y = x 3

3x 2


Table of values


+ 3.


-2\ - I \ - 0.15



- 15



+ 3.08


0 \ +1 \



2.1 5 \ + 3 - 3.08


Power functions with positive exponents Concept of a power function. A function y = x", in which n is an integer, is called a power function. If n is positive, the function is a polynomial, but if n is negative, n = -v (v > 0, an integer), then the function can be expressed in the form y = I/x' and is a rational function. The polynomial functions y = x' are even if their exponent n = 2m is even; they decrease monotonically for -00 < x .;;;; 0 and increase monotonically for 0';;;; x < +00. For odd exponents n = 2m + I the functions y = x' are odd; they increase monotonically everywhere. Even polynomial power functions y = x 2 ... The curves represented by these functions are symmetrical with respect to the y-axis, and their curvature is everywhere positive (Fig.). Each of them contains the origin (0, 0) and the points Q( -1 , + I) and P( + 1, + 1). In the neighbourhood of the vertex (0, 0) the tangents are the flatter the larger m is, whereas in a particular neighbourhood of the pointsQ and P they are the steeper, the larger m is. To every point (XI' YI) on Y = x 2 .. , a point (X2 , Y2) on Y = x 2"'(m2 > ml) can be determined by means of the differential calculus, so that the tangents at the two points are parallel. These curves are called parabolas of order 2m (Fig.).

..... :1 r- :i 1""

y ·:t IJ1

r '" /

~ '- r\ f-

" ","'

"" ~



":-r. f-\ -\






t Q~ I I









J -WI 1/ _


. '"'

""' I



l - f-


5.2-14 Graphs of the functions y = Xl," for m = 0, 1,2, ... ; y = x· is not defined for x = 0


Q7 Q6


a" Q3 02 a1

x a1 02 03 0." 05 a6 a7 aB 0.9


5.2-15 Portions of the curves of the functions y = Xl," as parabolas of order 2m

Odd polynomial power functions y = x 2 ..+1 . The curves are centrally symmetric about the origin. Except for the angle bisector of the quadrants I and III (y = x) they have negative curvature for all negative values of their domain of definition (-00 < x < 0) and positive curvature for positive values (0 x +(0), and so they have a point of inflection at the origin. Each of these parabolas of order 2m + 1 contains the points (+1, +1) and (-1, -I), and in the neighbour· hood of these points their tangents are the steeper, the greater m is; but in the neighbourhood of their common point of inflection (0, 0) the tangents are the flatter, the greater m is (Fig.).

< = log. (l /XX ) = -log. XX = -x log. x . For k = -I in particular, the graph of the function is the result of taking the mirror image in the +x-axis of the graph of the function y = log. x. Thefunctiony = -log. x = log. X-I = log. (l/x) is the inverse of the function y = a-X.


The function y = log. (kx). For positive values of the constant k, because y = log. (kx) = log. k + log. x, the graph of this function is obtained from that of YI = log. x by translation parallel to the +y-axis through a distance d = +Iog. k. For negative values x =. -k(k > 0) the function is defined only for negative values of x, and then because xx = Ikxl the function values are the same as those of y = log. (kx) with 0 < x < +00. Trigonometric and circular functions The trigonometric or angular junctions and the circular or arc junctions are a very frequently occurring type of irrational function . They are investigated more thoroughly in Chapter 10. Connections between trigonometric and circular functions. Because the circular functions are defined as inverse junctions of the trigonometric functions, the following connections are immediate: sin (arcsin x) = x; cos (arccos x) = x and so on. For positive x, if the principal value is taken on both sides, it is also true that Arceot x = Arctan (l/x), because cot x = I/tan x. For this reason the function y = arccot x can be dispensed with for most investigations. The following further relations are of interest: sin (Arccos x) = cos (Arctan x) =

(1 - Xl), sin (Arctan x)



+ Xl)



, tan (ArCSin x)





,cos (Arcsin x)



V(l - Xl),

(I _ Xl) , tan (Arccos x) =

(1 - Xl) x

It is enough here to give the justification for the first relation, because the others can be obtained in exactly the same way. From sin l y + cos 2 Y = I it follows that sin y = ± V(1 - cos l y), or

sin (Arccos x) = ± V(l - Xl), for the principal value of arccos x, 0 :;;;;; Arccos x :;;;;; n . In this domain of definition the sine function has no negative values, sin (Arccos x) ;;;. 0; the square root can therefore have only a positive sign, and sin (Arccos x) = + V(l - Xl), as given in Euler's formulae el CA. A triangle is called isosceles if two sides (a) are equal, the third side is called the base (c), the opposite vertex is called the apex . In an equilateral triangle all three sides are equal. A triangle is called acute if all interior angles are acute, right-angled if one interior angle is a right angle, and obtuse if one is obtuse. In a right-angled triangle the side opposite the right angle is called the hypotenuse (Fig.).


right- angled




7.4-2 Types of triangles

Basic facts about triangles Relations between the sides. From any vertex it is possible to reach any other vertex by traversing the sides in two different ways : either by going along the connecting side or along the other two, via the third vertex (for example, from A to B along c or along b and a via C) . Since the straight a + b, b a + c, and a b + c, line is the shortest distance between two points this implies that c from which six further inequalities can be deduced by subtraction: c - a b or a - c b, c- b a or b - c a, and b - a c or a - b c. But only three of these make sense geometrically, since the difference between two segments must have non-negative length. In a triang le the sum 0/ any t wo sides is greater than the third side. In a triangle a ny side is g reater than the difference between the other t wo sides.

b implies that a>fJ proved: if", p, then a b. In a triangle the angle opposite the longer of two sides is greater than the angle opposite the other. The side opposite the greater of two angles is longer than the side opposite the other. Angles opposite equal sides are equal and vice versa . Every isosceles triangle is symmetrical. The perpendicular from the apex to the base bisects the base and the angle at the apex. The angles at the base are equal. In a right-angled triangle the acute angles are complementary. In an isosceles right-angled triangle, the base angles are 45°. In an equilaterial triangle the interior angles are all equal; each is 60°. Equilateral triangles have three axes of symmetry. If one of the angles of a right-angled triangle is 300, then the opposite side is hal[ the hypotenuse.



The last theorem is a consequence of the symmetries of the equilaterial triangle and is frequently used; for instance, set-squares are usually right-angled triangles of this type or isosceles right-angled triangles. Congruence of triangles General remarks. Plane figures are called congruent if they are of the same shape and the same size. Congruent figures can be carried into each other by a transformation that moves points, but does not change incidence relations (between points and lines), angles between lines, and lengths of segments. Such a transformation also preserves areas and leaves parallel lines parallel. If congruent figures have the same orientation (with respect to some fixed orientation of the plane), they can be transformed into each other by a sequence of translations and rotations of the plane. Such figures are called directly congruent. If they do not have the same orientation, then a sequence can be found taking one into the other, which apart from successive translations and rotations has a single reflection in a straight line. Such figures are called inversely congruent. Translations, rotations, and reflections are called congruence transformations and can be used as criteria of congruences in the investigation of plane figures; but this by no means exhausts their usefulness as a tool of discovering new geometrical facts. Four theorems on congruence of triangles. In the definition of congruence it is required that the figures agree in all aspects, in particular, that the lengths of corresponding sides and the angles between them are equal. The theorems of this section state that for triangles in certain cases it is sufficient to test three parts as a check for congruence - if they are equal for two triangles, then the triangles are congruent. Here are the theorems.

7.4. Triangles


1. Two triangles are congruent if the length of a side of one is equal to the length of the corresponding side of the other and two angles of one are equal to the corresponding angles of the other (cr, I, cr). 2. Two triangles are congruent if the lengths of two sides of one are equal to the lengths of the corresponding sides of the other and the angles hetween these sides are equal (s, cr, I). 3. Two triangles are congruent if the lengths of two sides of one are equal to the lengths of the corresponding sides of the other and the angles opposite the larger sides are equal (s, I , cr). 4. Two triangles are congruent if the lengths of the three sides of one are equal to the lengths of the corresponding sides of the other (s, I , $). If one tries to construct triangles with three given sides and angles, one sees that if these correspond to one of the theorems, and only then - the triangle is uniquely determined. On the other hand, if two sides and the angle c = 5cm opposite the smaller side are given, say a = 3 a _ Zem units, c = 5 units, and IX = 20°, then the figure shows that there are two possible triangles with these measurements. For if a line is drawn at 20° to the segment IABI = c and an arc is drawn about B with radius a, then this A e 8 A c 8 arc intersects the line in two points C' and C". Both the triangles ABC' and ABC" satisfy 7.4-6 These triangles agree in three data, but are the requirements of the construction. Again, not congruent if IX = 80°, then the arc of radius a does not intersect the free arm of 0< at all, and there is no triangle that fits the requirements (Fig.). In constructions using the first congruence , " ", theorem it is convenient first to obtain the I/ a 80' I \~ two angles adjoining the given side. If one of them is not given in the data, it can be found A c\ 8 J by the theorem on the sum of the interior \, I ; angles of a triangle. For instance, if c, IX and y ~ , / are given, {J = 180° - IX - Y can be coma - 2cm puted. Then the triangle can be constructed 7.4-7 No triangle with these data can be constructed simply by drawing lines at the appropriate angles through the end-points of the given segment. Alternatively, the computation can be avoided by drawing a line under angle y to the free arm of 0< in an arbitrary point C'. The parallel to this line through B is then the third side of the triangle ABC.


- 4 sin 3 q:> . In the equation sin (q:> + !p) + sin (q:> - !p) = 2 sin q:> cos!p one puts", = q:> + !p, fJ = q:> - !p, so that q:> = 112('" + fJ),!p = 112(", - fJ),and one obtains sin", + sin fJ = 2 sin 112('" + fJ) cos 112('" - fJ). sin ", ± sin fJ sin", cos fJ ± cos'" sinfJ sin ('" ± fJ) tan", ± tan fJ = - - - - = = --'--,--,,"" cos'" cos fJ cos'" cos fJ cos '" cos fJ Sums, differences and products of trigonometric functions sin

+ sin fJ

+P - fJ = 2 sin - 2 - cos - 2 -


+ cos P = 2 cos - 2 - cos - 2 -

", + P

- P


• fJ 2 + {J.-fJ - sm = cos - 2 - sm - 2 -


· +p . - cos P= - 2 sm-2- s m -2-


sin ( + fJ) +....:.·f3)-icot + cot fJ = ...:.s..,;. + tanfJ = -....:.......:.....!...,;sin sin fJ cos cos fJ sin ( - P) cot _ cot fJ = - sin ( - fJ) tan - tan {J = ---'---'--i.sin sin {J cos cosp cos + sin = V2sin (450 + ) = V2cos (45 0 ) cos - sin = V2cos (45 0 + ) = V2sin (45 0 ) sin ( + f3) sin ( - P) = cos 2 P- cos 2 cos ( P) cos ( - P) = cos 2 P- sin 2 sin sinp = 1/2[cos( - P)-cos( + fJ») cos COSP = I / 2 [COS ( - f3) + cos( + {J)I sin ex cos P = 111[sin (0: - P) + sin (0: + P)I cos 0: sin fJ = 1/2[sin (0: + (J) - sin (0: - P») tan + tan P tan - tan fJ {J cot + cot P cot - cot P tan tan fJ = cot + cot {J = - cot ", _ cot {J cot cot = tan + tan{J tan - tanp tan


cot {J = tan + cot {J = _ tan - cot fJ cot + tan {J cot - tan P sin sin{Jsin y=1 /4[sin( + {J - y) + sin({J + y - ) + sin(y + -f3) - sin( + {J + y)1 cos o: cos{Jcos y = 1/ 4[cos ( + P - y) + cos ({J + y - ) + cos (y + - f3) + cos ('" + (J + y) ] sin sin {Jcos y = 1/4[- cos( + fJ - y) + cos({J + y - ) + cos(y + - {J) - cos( + (J + y) ] sin", cos{Jcosy = 1/ 4[sin ( + (J - y) - sin ({J + y - ) + sin (y + - fJ) + sin ( + (J + y) ]


Powers of trigonometric functions sin 2 q:> = 1/ 2(1 - cos 2q:» sin 3 q:> = 1 / .. (3 sin q:> - sin 3q» sin· q:> = l/ a(cos 4q:> - 4 cos 2q:> + 3) sin' q:> = 1/ 16(10 sin q:> - 5 sin 3q> + sin 5q:»

cos 2 q:> cos l q:> cos" q:> cos' q:>

= 112(1 + cos 2q:» = 1 / .. (3 cos q:> + cos 3q» = I/ a(cos 4q:> + 4 cos 2q:> + 3) = 1/ 16(10 cos q:> + 5 cos 3q> + cos 5q:»

General formulae for the sine and cosine of a multiple angle. De Moivre's theorem in the theory of complex numbers states that (cos q:> + i sin q:»" = cos nq:> + i sin nq:>. Bearing in mind that i 2 = -1 this can be proved for n = I, 2, 3, ... by the method of induction by means of the addition theorems. If the left-hand side is expanded by the binomial theorem, equating the real and imaginary parts gives cosnrp = cos'rp - ( ; ) cos·- l q:> sin l q:> + ( : ) C05· - " rp sin" rp - .•.

sin nq:>

= (; ) cos·- I rp sin rp -

( ; ) cos·- 3 q:> sin l q:>

+ (~) cos·- ' q:> sin' rp -



10. Trigonometry

'The geaeraI siDe curve. In nature and technology the mathematical description· of oscillations, for example, in high frequency technology, optics, acoustics or mechanics, is based on sine and cosine functions. In these oscillations the greatest displacement, the amplitude a, of a sine oscillation can be different from 1, its wave length A different from 2n, and the ordinate at the :uro point different from o. The function Y = a sin x, for example, has the amplitude a (Fig.) and the function y = sin (2nx/A) the wave length A, because for 0";;; x ,.;;; A the argument 2nx/A runs through the values from 0 to 2n (Fig.). The function y = sin (nx), where n is an integer, has exactly n complete oscillations in the interval from 0 to 2n, since A = 2n/n. Finally, the function y = sin (nnxf/) with A =21/n describes an oscillation of which n waves have length 21.


10.1-27 Graphs of the functions y and y = ',. sin x

sin x, y



4 sin x


10.1-28 Graphs of the functions y




and y


sin (nx/1O)

Superposition. If several physical quantities that can be represented by oscillations act at a point, then the ordinates for this point are added. For example, Yl = 2 sin x and Yl = -cos 2x gives Y = Yl + Y2 = 2 sin x - cos 2x (Fig.). :I y"y,'h" 2sinx-cos2x


10.1-29 Graph of the function y

= y,

+ y. =

2 sin x - cos 2x

Damped oscillations. If an oscillating system loses energy, then the amplitude decreases. For example, the function a = 3e- 2x ,,, has the value 3/e for x = n/2, only 3/el for x = 2n/2, and so on. The figure shows the graph for Y = 3e- lx ,,, sin 4x. Angular frequency w and phase difference rp. If the time t is regarded as the independent variable, the equation of the general sine curve has the form Y = a sin (wt + rp). From the fact that for wt = 2n a whole oscillation is completed it follows that the time for a complete oscillation (through wave peak and wave trough) is t = 2n/w. This time is called the periodic time of the oscillation and is denoted by T. If T is measured in seconds, then 1/T is the number of oscillations in one second,

10.2. Trigonometric equations


that is, the frequency f of the oscillation: f = lIT. The angular frequency co = 2nIT = 2n(l/n = 2nf gives the number of oscillations in 2n seconds. Finally, the phase difference tp is the angle by which the given curve leads the sine curve (Fig.). For t = 0 the function y already has the value y = sin tp. For a negative phase difference tp one speaks of lagging. For a = 1 and tp = +n/2 the function y = a sin (cot + tp) becomes the cosine function cos cot, that is, the cosine curve leads the sine curve by n/2. If a general sinusoidal oscillation with angular frequency co is given, then a and tp


o 10.1-30 Graph of the function -1

y = 3e -2,,/,. sin 4x

10.1-31 General sine curve + ,,). Left, -2 phaser diagram or vector diagram; right, curve repre· sentation or line diagram -J y y = a sin (wt


can be characterized in a phaser diagram (vector diagram) in which a is the radius of the circle from which the sine curve can be constructed and tp is the angle between the phaser (vector) and the positive abscissa axis at the time t = 0 (Fig.).

10.2. Trigonometric equations The expressions considered so far have been algebraic in T (see Chapter 4.). The notion of an expression will now be generalized so as to include sin T, cos T, tan T and cot T. By equating expressions and at the same time taking into account the range of values of the variables, new equations are formed . In trigonometric equations with one variable, the variable x occurs in at least one such generalized expression. In pure trigonometric equations x occurs only in such expressions, for example, in sin (2x + n) - V2 cos x = 0; in mixed trigonometric equations x also occurs in algebraic expressions, for example, in tan x - 3x = O. Trigonometric equations are transcendental equations (see Chapter 4.). There is no general algorithm for their solution, but they can be solved graphically, or by numerical approximation methods, with arbitrary precision. For certain special types of pure trigonometric equations solution algorithms do exist. Because of the periodicity of the trigonometric functions the domain of the variable of a trigonometric equation is often confined to an interval whose length is a primitive period, say 0 ~ x 2n.

12): cos (3xf7) + cos (n12 - x) = O. r-~~~~~~------------------,


5x/ 7

= 3n/4 + kn,

Since k can take the values 0, ± I, ± 2, ... one can replace - k by + k in the formula for XI: XI = - 7n/ 8 + 7kn / 2; X2 = - 2In/20 + 7kn/ 5 . Tests show that all the values satisfy the equation . It should be noted that the solutions for consecutive integers k differ not by 211:, but by 7n/2 or 7n15, respectively (Fig.). y 1

10.2-4 Intersections of the graphs of the functions y, = cos (3x/7) and Yo = -sin X, the

belong to


these marked black to



10. Trigonometry MIxed trigoaometric equatioDs

Mixed trigonometric equations can be solved only by graphical or iterative methods (see Chap..

ter 29.).

Example J: The solutions of the equation cos x - xl2 + 1.7 = 0 are the abscissa.e of the poinls of intersection of the curves with equations YI = cos x, Yl = x/2 - 1.7 (Fig.). They have only

10.2-S Graphical solution of the equation cos x = x/2 - 1.7 one point of intersection with the abscissa Xo R:4 2.21. If the graphs in the ne.ighbourhood of this intersection are drawn on a larger scale, the accuracy of the reading can be improved. Here one obtains Xo R:4 2.209. Test: cos 2.209 -

2.~09 + 1.7 = cos 140.63' + 0.5955 = -


+ 0.5955 = -


A closer approximation XI to the correct value is given by Newton's method for approximate solutions; XI = Xo - f(xo)/f'(xo>. f(xo) = cos Xo - xo/ 2 + 1.7 = - 0.0003, !'(xo) = - sin Xo - 1/2 = - 1.3032, XI = 2.2088 . The approximation can be further improved by the repeat.ed application of Newton's method. Example 2: The graphical solution of the equation 3 tan X - 2x = 0 by means of the functions YI = tan x, Y:z = 2x/3, yields the solutions Xl = 0, X2 = ± 4.38, X3 = ± 7.65, ... For increasing values of X the solutions approach more and more closely tbe odd multiples of n/2. To every solution Xo there corresponds the equal and opposite solution - Xo ; for tan Xo = be o/3 a.lso implies that tan (- xo) = 2!J(_ XO) (Fig.).

10.2-6 Graphical solution of the equation 3 tan x - be



11.1. Solution of right-angled triangles


11. Plane trigonometry 11.1. Solution of right-angled triangles .. 241 General methods . ...............• 241 Applications. . . . . . . . . . . . . . . . . . . .. 242 11.2. The trigonometric functions in the general triangle. . . . . . . . . . . . . . . . .. 244 The formulae of plane trigonometry. 244 The four main cases in the solution of a triangle . . . . . . . . . . . . . . . . . . . . . .. 246

11.3. Further formulae and applications . 248 Geometry ....... . ......... ...... 248 Physics. . . . . . . . . . . . . . . . . . . . . . . . . 250 Technology . .............. . . . .... 251 Navigation . . . . . . . . . . . . . . . . . . . . . . 252 Trigonometric determination of heights ......... . ............. . . 253 Surveying. . . . . . . . . . . . . . . . . . . . . .. 255

The trigonometric functions already defined make it possible to use angles to calculate unknown quantities in plane rectilinear figures. Angles can often be measured with less effort and greater accuracy than lengths. As the name indicates, trigonometry is concerned with the measurement or calculation of triangles into which every figure bounded by straight lines can be subdivided by diagonals. In this one always has in mind the use of known angles.

11.1. Solution of right-angled triangles General methods

The definition of the trigonometric functions was first given in the right-angled triangle and then extended to arbitrary angles with the help of the unit circle. These definitions contain all the relations between lengths and angles in the right-angled triangle and thus suffice to calculate all the rest when any two of the six quantities are given. When the right angle is denoted by y and the hypotenuse by c, two additional relationships in the right-angled triangle ABC (Fig.) are available from geometry: I. the theorem of Pythagoras: c 2 = a2 + b2 , II. the fact that each of the angles with its vertex on the hypotenuse is the complement of the other: IX + P= 90°. From these relationships or by re-Iettering the triangle all possible cases in which two of the quantities a, b, c, IX and P are given can be reduced to four cases, namely c, IX; c, a; a, IX and a, b, for which the solutions will now be stated. 1. Given the hypotenuse c and one adjaceht angle, say IX: 1. P = 90° - IX; 2. sin IX = a/c, a = c sin IX; 3. coS IX = b/c, b = c cos IX. II. Given the hypotenuse c and one other side, say a: 1. sin IX = a/c; 2. P = 90° -IX; 3a. b = V(c 2 - a2 ) or with the help of the calculated angle IX: 3b. cot IX = bfa, b = a cot IX; or 3c. cos IX = b/c, b = c cos IX. 11.1-1 Right-angled triangle III. Given an angle and the side opposite to it, say a and IX: l.P=900-1X; 2. cot IX = b/a, b=acotlX; 3. sin IX = a/c, c = a/sin IX or with the help of the calculated angle p: 2a. tan p = bfa, b = a tan p; 3a. cos p = a/c, c = a/cos p. IV. Given the two sides a and b containing the right angle : 1. tan IX = alb, 2. p = 90° - IX; 3a. c = V(a 2 + b2 ) or with the help of the calculated angle IX: 3b. c = a/sin IX; or 3c. c = b/cos IX. Checks and accuracy. One usually tries to find the solution using only the given quantities.

Auxiliary solutions with the help of quantities already calculated can be used as checks, because

the same quantity calculated in different ways must theoretically have the same value. Another

check is based on the theorem that the sum of the angles of a triangle is 180°. In surveying checks

are provided for almost every trigonometric calculation. In this the permissible deviation of the value for the same quantity depends essentially on the tables used. In evaluating a possible deviation

11. Plane trigonometry

242 y



y.tan9'1 1 1




one must bear in mind that for a given small interval Lli P of an angle p the errors Lly in looking up values of different trigonometric functions are of different magnitude. In the figure, for example, Ll 3 y for y = tan p is greater than LIlY for y = cos p. Of course, conversely, for a given small interval Llzy the value of the angle can be determined more accurately from the tangent function or from the cotangent function than from the other two functions. For the function y = sin p, in particular, the figure shows once more the dependence of the magnitude Lly of the interval of the function values upon the magnitude of the interval of the angle values. For small values of



11.1-3 Inclination of a ladder leaning against a wall, h = 1.2, / = U

11.1-2 Accuracy in working with trigonometric functions

the angle in the neighbourhood of p = 0°, Lly is large; on the other hand, for large values in the neighbourhood of p = 90°, Lly is small. The angle p can be determined from the value found for the sine with greater precision in the first case than in the second. The accuracy of the check must, of course, be in agreement with the measured value. To calculate the angle made with the horizontal by a ladder of length I = 1.50 m leaning against a vertical wall at a height h = 1.20 m, one obtains sin p = 1.2/1.5 = 0.8 (Fig.). The distance x of the foot of the ladder from the wall is given by x = V{(1.5)Z - (1.2)Z} = 0.90 m. As a check x. = 1.5 cos P and x, = 1.2 cot P are calculated. The round value PI = 53° taken from a 4-figure table without interpolation gives the values Xis = 0.903 and XI, = 0.904 which correspond to the accuracy of I and h. From a 7-figure table one obtains the less meaningful values pz = 53°7'48.4", X2. = 0.9000000 and X2, = 0.8999996. The distance of the ladder will hardly be measured to within 4 millimetres and certainly not to within 4 ten-thousandths of a millimetre. The result cannot be more accurate than the given values. To increase the accuracy in surveying, additional measurements are made and the most probable value is calculated by the methods of errors and least squares. Applications Length of 8 chord of a circ:le. The angle subtended at the centre of a circle of radius r by the chord of length s is twice the angle subtended at the circumference by the same chord (Fig.). The perpendicular from the centre M of the circle to the chord s bisects both the angle at the centre and the chord and forms two congruent right-angled triangles. It then follows that: sin y = s/2r or s = 2r sin y. a Oo-----~~_.----------_r----~.

11.1-4 Chord of a circle


11.1-S Determination of a right angle from a hidden point

11.1. Solution of right-angled triangles


Determination of a right angle from a hidden point. From a water pipe running in a straight line between the villages D and E (Fig.) a perpendicular branch pipe to a village N ist to be constructed and a water tower is to be built on the intervening ridge. N cannot be seen from the required point Fat which the branch pipe leaves the main pipe, though it can be seen from D and E. The distance a = DEI and the angle 0 are measured. The position of F on DE is determined by the distance x = DF . From the right-angled triangles DFNand EFNoneobtains: IFNI = x tan 0, IFNi = (a-x)tan6, tan E so that x tan 0 = (a - x) tan 6 and hence x(tan 0 + tan 6) = a tan 6, x = a ----;~=:..::..-.-­ tanu + tan 6 For the calculation of x using logarithms this expression is transformed using the addition theorem: a sin 61cos 6 a sin 6 cos 15 cos 6 cos 15 sin 6 x = sin t5lcos 15 + sin 61cos 6 = cos 6(sin 15 cose + cos 0 sin 6) = a sin (0 + 6) . Determination of heights. The height of a tree can be determined (Fig.) by measuring the angle of elevation of the top of the tree from a point A, the distance s between the foot of the tree F and the base S of the point of observation, and the height h2 of the measuring instrument (that is, the vertical distance IASI). Then hl = s tan'll, and the actual height H of the tree is given by H = hl + h z = s tan'll + h 2 • Approximate methods of determining heights. 1. Instead of measuring the angle of elevation'll, the top of the tree can be sighted along the hypotenuse of an isosceles right-angled triangle ABC in which the side CB is held in a vertical line by a plumb line. The angle", is then 45° and hl = S, H= s+ h z .


A -~S:---.L...--J---......J"-I F '--L.

11.1-7 Method of measuring heights in forestry

This method can be used only when there is enough room to choose the point of observation suitably. Otherwise one can employ the following method, which is usual in forestry. 2. A rectangle ABCD (made of wood or cardboard) is held in such a position that the top G of the tree is sighted along the edge AB (Fig.). A plumb line suspended from the point B then cuts the side CD of the rectangle in the point L. The two angles marked 6 are equal, since the arms of one are perpendicular to the corresponding arms of the other and the right-angled triangles BCL and BEG are similar. Then IGEI/IBEI = tan 6 = ICLI/IBq If one chooses IBCI = 10" and subdivides the side ICDI into inches, then ICLI/IBCI = ICLI/IO is always a decimal fraction whose value is tan E. The rectangle ABCD •calibrated' in this way is a disguised table of tangents, which is particularly simple to handle. From hl = IGEI = stan 6 it follows that the height of the tree H = stan 6 + h2 = s(ICLI/IO)

+ hz ·

Determination of the altitude of the SUD. From the length b of the shadow cast by a vertical rod of length s on a horizontal plane (Fig.) the angle rp between the rays of the sun and the horizontal can be determined. It is called the altitude of the sun. One obtains tan rp = sib or cot rp = b/s. If the rod is of length 1 yard, then the length of the shadow in yards gives the value of cot rp immediately. The angle of a tip. If sand is transported on a conveyor belt, then a conical heap or a sand tip is formed as it falls off (Fig.). Its content can be calculated from the diameter d = 2r of its circular base -/,~.:--:;-....,c.-:.and the tip angle IX between a line in the curved 11.1-8 Altitude of the sun 11.1-9 Sand tip surface of the cone and the


11. Plane trigonometry

horizontal. V = nr zh/3, where h = r tan 0:, so that V = (nr 3 /3) tan 0:. If the vertical angle y of the cone is used instead of the tip angle 0:, then h = r cot (y/2) and V = (nr 3 /3) cot (y/2). For sand the tip angle is approximately 33° and for vulcanite about 36°. The angle between the plane faces of a regular tetrahedron and a regular octahedron. The regular tetrahedron is bounded by four congruent equilateral triangles and six edges of equal length k. The angle to between two adjacent triangular faces can be seen in a plane section of the tetrahedron containing the edge BD, bisecting the edge AC skew to BD, and perpendicular to AC (Fig.). The section BDM is an isosceles triangle. Its equal sides are altitudes of faces of the tetrahedron and have length h = l/zk V3. The height 1J of the tetrahedron is perpendicular to one of these equal sides and divides it in the ration IMFI: IFBI = 1 : 2, because the altitudes of the equilateral triangle ABC are also medians. In the right-angled triangle MFD, h is the hypotenuse and IMFI = h/3 the side adjacent to the angle v. Hence cos to = 1/3h/h = 1/3, to = 70°31'44". The regular octahedron is bound£ ed by eight congruent equilateral o triangles and twelve edges of equal length k. The angle 21' between two adjacent triangular faces can be seen in a plane section through c two opposite vertices E, F and through the midpoints Mit Mz of two parallel edges (AD II BC) that are skew to the line EF joining these vertices (Fig.). The section is a rhombus of side h = 1/2k V3 A whose diagonals, IEFI = k V2 and F IM1MzI = k, bisect the angles of 11.1-11 Octahedron 11.1-10 Tetrahedron the rhombus and are at right angles to one another. Hence from the right-angled triangle M 1 GE it follows that the half-angle l' is given by: cosp. = 1/2kW/2k V3) = I/V3 = 1/3 V3; P. = 54°44'07" or 2p. = 109°28'14".

11.2. The trigonometric functions fu the general triangle In many cases the lengths and angles accessible for measurement do not lie in right-angled triangles. Relationships between the sides and angles of the general triangle were therefore derived. The most important are the sine rule and the ccsine rule. They are sufficient for every calculation. The cosine rule is less advantageous for calculations, especially when tables are used, because the formula contains a sum of squares and a product term. It can be replaced by the tangent or by the half-angle formula. The formulae of plane trigonometry The sine rule. Every triangle ABC (Fig.) has a circumcircle whose centre M is at the intersection

of the perpendicular bisectors of the sides of the triangles. The sides of the triangle are chords of he circle and the opposite angles are angles at its circumference. If the radius of the circumcircle is denoted by R, then the sides can be calculated as chords of the circle: a = 2R sin (x, b = 2R sin p, c = 2R sin y. From these one obtains for the diameter 2R = a sin 0: = b/sin p = c/sin y.


Sine rule a/sin 0:



b/sin p = c/sin y or a : b : c = sin 0: : sin p : sin y

In an)' triangle the ratio of each side to the sine of the opposite angle ii a constant (equal to the diameter of the circumcircle). The sine rule. In a plane triangle the ratio of any two sides is equal to the ratio of the sines of the opposite angles.

The sine rule connects opposite data. If two opposite data are given, then from any third datum one can calculate the opposite one. Given a, (X and b, for example, p can be determined from sin p/sin (X = bfa, sin p = (b/a) sin 0:; or given b, p and y the side c can be determined from c/b = sm y/sin p, c = b sin y/sin p.


The sine rule

11.2. The trigonometric functions in the general triangle


In calculating an angle by means of the sine rule one should, of course, observe that two angles and IP2 are given by sin IP, as can be seen from the unit circle. One of these angles is acute and the other is the difference between the acute angle and 180°; IPI + IP2 = 180°. One must distinguish in each particular case which of these angles corresponds to the given geometrical situation.


The cosine rule. In the triangle ABC let D be the foot of the altitude he and IADI = q the projection of the side b on the side e (Fig.). This projection q = b cos ex is positive for an acute angle ex and negative for an obtuse angle. The segment DB ist thus of length e - q = IDBI for arbitrary values of ex. The altitude he always has the length he = b sin ex. Applying the theorem of Pythagoras to the right-angled triangle DBC one obtains a 2 = h: + (e - q)2 = b 2 sin2 ex + e2 + b 2 cos 2 ex - 2eb cos ex, or a2 = b 2 + e 2 - 2be cos ex. Corresponding relationships can be found using the altitudes ha and hb • These can be obtained formally by a cyclic permutation in which a is replaced by b, b by e and e by a; the same holds for the angles ex -+ p -+ l' -+ ex.



.0 C)


b a



) ) .2-2 The cosine rule: a) for an acute-angled triangle, b) for an obtuse-angled triangle, c) cyclic permutation

Tbe cosine rule. In a plane triangle the aquare of ODe side Is equal to the sum of the IQ1IIlftS of the other two sides min... twice the product of these two sides and the cosine of the angle betweea diem. When two sides and the included angle are known, the third side can be calculated using the cosine rule, and when three sides are known any angle can be found:

cos ex =



cos P =



,cos l' =



The tangent formula. Using the rule for the ratios of corresponding sums and differences and applying the addition theorems one can deduce: a sin ex a- b sin ex - sin p 2 cos [(ex + P)/2) sin [(ex - P)/2) b = sin P' a + b = sin ex + sin P 2 sin [(ex + P)/2) cos [(ex - P)/2) Dividing both numerator and denominator by cos [(ex + P)/2) cos [(ex - P)/2] one obtains the tangent formula for the sides a and b. The corresponding formulae for the remaining pairs of sides are obtained by a cyclic permutation: a - b tan [(ex - P>/2] b- c tan [



III. Given two sides and the included angle. The solution comes from the cosine rule or th,. tangent formula. Given the values of b, c and tX in the triangle ABC, then the cosine rule gives a2 = b2 + c 2 - 2bc cos tX and from this the unique value a = V(b 2 + c 2 - 2bc cos tX). The angle P can also be determined uniquely from the cosine rule, that is, from cos P = (c 2 + a 2 - b 2 )/(2ca). However, it is usually preferable to use the sine rule and obtain sin P = (b/a) sin tX . Of the two angles PI and P2 that satisfy this equation only one corresponds to the geometric conditions. From (y + P}/2 = 90° - tX/2, and by the tangent formula one obtains: tan [(y + P/2) (c - b)/(c + b) = tan [(y - P)/2) ; from (y + P}/2 and (y - P)/2 the angles P and y can be found. The third side can then be determined by the sine rule; c = a sin y/sin lX. Example: A cable is to be laid in a straight line through wooded country between two places Rand S. They are not visible from one another, but a point A can be fouod from which the distances d = IARI = 2.473 miles and e = IASI = 3.752 miles and the angle T = >

> >


11. Plane trigonometry


IV. Given three sides. The solution comes from the cosine rule or the half-angle formulae, that . . . b2 + C2 - a 2 IX (s - b) (s - C)] IS, from either of the equatIOns cos IX = 2bc ' tan "2 = s(s _ a) , and the equations obtained from these by cyclic permutation. Both solutions are unique and are obtained either from suitable combinations of the six numbers a2, bl , cl , 'lab, 2bc, 2ca or of the four numbers s, s - a, s - b, s - c. Therefore each of the three angles IX, fl,,, should be calculated and the value of the sum of the angles of the triangle used as a check. Example: Three points R I , R 2 • R J on raised ground are to be connected by radar (Fig.). At what angles must the transmitter and receiver at each point R I , R l , R J be built? IRIRz l = c = 45.21 miles; IRzRJ I = a = 52.46 miles; jR3R d = b = 39.37 miles. Half-angle formula

Cosine rule

a1 = 2752.0516

+c + a2 a2 + b

bZ c1

2 _

1 -

b1 = 1549.9969 c 1 = 2043.9441 a2 = 841 .8894 b2 = 3245.9988 c2 = 2258.1044 = 76°19'12" fl = 46°49'06" 56°51'42" 180°00'00"



11 .2-6 Solution of a lrianlle. liven three sides

a = 52.46

s - a = 16.06 t

39.37 s - b = 29.15 45.21 s - c = 23.31 2s - 137.04 - - s - 68.52 b




1.20575 1.46464 1.36754 1.83582

oc = 76°19'12"

fl = 46°49'06"


= 56°51'42" 180°00'00"

11.3. Further formulae and applications In many fields arguments are made precise with the aid of mathematical relations; for example, when directions and angles in plane rectilinear figures occur, then theorems of plane trigonometry are used. One of these fields, namely surveying, plays a special role. In this discipline the relationships in question rest more directly on these theorems than in other fields, and historically the requirements of surveying were responsible for the development of plane trigonometry. For this reason the possible applications in this field are dealt with in a special section.


Geometry The radius r of the inscribed circle. In a triangle ABC the bisectors of the angles intersect at the centre M of the inscribed circle. If one draws the radii through the points of contact E, F, G of the sides of the triangle (Fig.), then six right-angled triangles are formed. They are congruent in pairs and, in particular, the pairs of sides marked x, y, z, respectively, are equal. Their lengths are · x =s-a, y = s - b, Z = s - c, where s = (a + b + c)/2. In the triangle AGM, tan (IX/2) = r/x = ,/(s - a), but by the tangent formula for the whole triIX _ (s - b) (s - C)] Hence angIe tan T S(S _ a) .


_,_ =

v[ s(s - a) V[ (s - b) (s -

s- a , = (s - a)

(s -

b) (s -c) ] ,

s(s -


C)] .

The same result would have been obtained by considering tan ({J/2) or tan (,,/2).



11.3-1 Inscribed circle of a triangle

Radius of the inscribed circle



V[ ($ -

a) ($


b) (s -

c) ]


L....._ _ _ _ _ _ _.L..._..:......:'--_ _ _ _ _ _-=---..I

Marking out an arc of a circle wbose centre is inaccessible. Between two points A and B, whose distance apart e is known, arbitrarily many points P, are to be constructed, all lying on a circle through A and B with given radius, (Fig.). The centre of the circle is inaccessible. It is required

11.3. Further formulae and applications


to find the distance s from A of the points P, and the angle Vi between AP, and AB. Let P be one of the required points. Then 8 the triangle AMP is isosceles with base s = 2r sin (a/2) subtending an angle a at the centre of the circle. The angle 0 also. But this means that (IX - fl) > 0 or IX> fl.


1. The sum of two sides is greater than the third. The difference between two sides is smaller thall the third. Corresponding to each spherical triangle there exists a solid angle. This degenerates into a plane circular sector when the sum of two sides is equal to the third side, and is impossible in space if the sum is smaller than the third side. If the difference between the two sides a and b is greater than or equal to the third side c, a - b;;;. c, then it would follow that a;;;' b + c, in contradiction to the first part of the theorem. 4. The sum of two angles is less than the third increased by:'l (or J800). As has just been shown, in the polar triangle ABC: a + 1} e and a - 1} e. Because a = 180° - IX, 1} = 180° - fl, e = 180° - Y this means that for the triangle ABC: 180° - IX + 180° - fl > 180° - y, and 180° - IX - 180° + fl 180° - y, 180° + y > IX + {J, and {J + y 180° + IX .


9S .5S0

48.2S· 47.30·

y/2 y/2

z = 90° + 62 - h 2 • that is, q;2 = 44.79°. ($ -

12.3-11 Schematic representation for Example 2 of the sky (left) and of the earth's surface (right), Q equator, mo meridian of Greenwich The two observation points PI and P z , togetber with the Nortb Pole N determine a spherical triangle PINP1 on the earth's surface. With a course angle (X = 67.5° at the point PI. the ship has travelled a distance P";'P2 = 15.2 nautical miles = 15.2·1.852 kID between the observation . . pomts, an d t h·IS correspon d s to an arc s = 360·15.2·1.852 2nR = 02053° . . Tbe 51·de P'-'N '- oPPOsite the course angle

is 90° - q;2 = 405.21 0; the sine rule. sin LU = sin s sin (X/sin (90° - q>2), gives

282 LI)'

13. Analytic geometry of tbe plane

= 0.329°. In the same triangle Napier's analogy 2a) gives: tan [(90° - 1111)/2] = tan [(90° - IIIl - s)/2] sin [( + LI).)/2]/sin [(

- LlA)/2] , and hence 90° - III, = 4S.3°, 1111 = 44.7°. In the nautical triangle ZPNS I of the first observation point the three sides ZSI = 90° - hi, ZPN = 90° - 1111 and P;SI = 90° - b l are known . By the cosine rule for sides the difference t ' between 360° and the hour angle t can be calculated : cos t ' = (sin h, - sin IIIl..Sin b,)/(cos III, cos b,) . One obtains t ' = 4S.)3° = 3.01 h = 3 h 0 min 36 s. At the first observation it was l2h - 3homln 36' = 8hS9 m 'n24' true local time, or 8h44 mlo21' local mean time, since t", = t, - E.T. Relative to the local mean time of Greenwich the time difference is 18b Som in - 8h 44 mln 2l ' = 10 h OS min 39 s, or 10.094 h. Consequently the difference in longitude is 10.094' IS° = ISI.41 °. Thus, Greenwich lies east of P, and the longitude of P, is)., = 151.41 ° Wand that of P 1 is Al = A, + Lll = ISI.74°.

13. Analytic geometry of the plane 13.1.

Plane coordinate systems ...... . .. 282 Parallel coordinate systems . .. . . .. , 283 Polar coordinates . . . . . . . . . . . . . . .. 284 Changing from one coordinate system to another ........ . ............. 284

13.2. Point and line ................... 286 Segment and ratio of division . . ... . 286 Equations of a line . . . . . . . . . . . . . . . 287 Incidence of point and line . ... . ... 292 13.3. Several lines .. .... .. .... .. .... .. 292 Point and angle of intersection . . . . . 293 Triangle and polygon .......... .. . 295 13.4. The circle ...... . . . ........... . . 299 Equations of a circle . . ... . . . .. ... 299

Circle and line ......... . . .. .. . .. 300 Two circles .. .. . . . . .. . . ..... .... 302 13.S.

The conics. . . . . . . . . . . . . . . . . . . . .. 302 Conics as intersections of a circular cone with planes ............ . .. . . Equations of the parabola ......... Equations of the ellipse . . .... ..... Equations of the hyperbola .. . ..... Conic and line .... . . . .. .. .. . .. . . Normal and polar of a conic. . . . . .. Two conics .... . ......... . ... . .. Common vertex equation of the conics Polar equations of the conics .. . .. . Discussion of the general equation of the second degree . .. . ........ . . ..

302 304

305 307 309 311 313


316 318

The main idea of analytic geometry is that geometric investigations can be carried out by means of algebraic calculations. This method has proved extraordinarily fruitful. The fusion of geometric and algebraic thinking, together with functional thinking, provides an important help to man's understanding of the exploration and comprehension of objective reality. At the same time the method is particularly attractive mathematically and gives rise to important elements in the training of the mind. The birth of the method of analytic geometry, and the consequent growth ofthe methods of the differential and integral calculus, characterize the transition to modern mathematics. The year of birth can be taken to be 1637, when DESCARTES (IS96-1650) published his Discours de la Methode anonymously, to avoid a dispute with the church. In this work, which is also significant for the history of philosophy, the third part, entitled La Geometrie, systematically expounds the fundamental principle of analytic geometry. Shortly before, FERMAT (1601-1665) had also worked out the method of analytic geometry, but his treatise Ad locos pianos et solidos isagoge (Introduction to planar and spatial geometric loci) was not published until 1679. Since the 'Geometry' of Descartes had also the better notation, the development of the method of analytic geometry is usually attributed to Descartes. Its present form was, however, developed a long time after Descartes, particularly by EULER (1707-1783). For example, DESCARTES did not use two axes, and only since the time of Euler, to whom a large part of the modern notation is due, have farreaching conclusions been drawn from the equations of geometric loci, while DESCARTES and FERMAT generally regarded their investigations as ending when the equation had been set up.

13.1. Plane coordinate systems The fusion of geometric and algebraic thinking is attained by regarding geometric figures as sets of points and by assigning numerical quantities to each point, which distinguish it from other points. A curve or a line is then the carrier of a totality of points whose numerical quantities satisfy

13.1. Plane coordinate systems


certain relations, which are called the equations of the figure, for example, the equation of an ellipse or a line. The graph of a linear equation in two variables is always a line, and that. of a quadratic equation is a conic. The foundation of this construction of analytic geometry is the correspondence between points and numbers, which must be one-to-one. On a line, or more generally a curve, one number is sufficient to fix a point uniquely, on a plane or a surface a number pair, in space a number triple; conversely, a point on a curve uniquely determines one number, on a surface a number pair and in space a number triple. These numbers are called coordinates. They can be obtained in different ways; coordinate systems are the means of fixing them. The number line. On a line the position of any point P is uniquely determined if a zero point 0 and a unit segment u = 01 are given on it. The integral multiples of the unit segment are obtained by repeatedly laying off u either from 0 beyond 1 in the positive direction or from 1 beyond 0 in the negative direction (Fig.). The end-points of the multiples correspond to the whole numbers, positive or negative. The point P is either an end-point, or it lies between two of the end-points, say nand n + 1; there is always a real number x such that x times u is the distance 10PI of the point P from the origin O. One has n';;;;; x ,;;;;; n + 1 for positive x, and -n';;;' x ;;;. - n' - 1 for negative x . The number x is the coordinate of the point P. Conversely, any posirive direction negative direction real number x uniquely determines a point p p P of the number line by means of the oI • I equation m(OP) = xu, where m(OP) -3 -2 -1 z 3 n n+1 = 10PI if x> 0 and m(OP) = - 10PI 13.1-1 The number line if x . _" intersect? - The following scheme shows the solution of the system of equations. ,-





I - I


. I·"!J""

+ j

" "",V




+ 3yo - 6 = 0 - -5xo + 15 = + 3yo + 9 = 0 +-" Xo = 3) + 3yo - 6 = 0 I

- 3X O 2xo

- 3(-
















4#" ' "",

*"" ,i"


13.3. 1 Determination of the point of in tersec tion of two lines

- It

The lines intersect at the point poe - 3, - I) (Fig.). 2. To find the point of intersection Po of two lines given in the nonnal form, say y = - 3x + 14, y = - x - I, one can most conveniently solve the system by equating the right-hand sides. In the given example, the lines intersect at Po{7.5, - 8.5). 3. The lines 3x + y - 7 = 0 and 2x - y - 3 = 0 intersect at the point Po{2, I). 4. The lines 2x - 3y + 5 = 0 and 3y - 2x + 2 = 0 are parallel. For each line the origin belongs to the positive half-plane. In the Hessian normal form the system of equations becomes

+ +



+ +


2xo - 3yo 5= + 2xoIV I3 - 3Yo/V 13 5/V13 = 2xo 3yo 2= 0 - 2xoIV I3 3YoIVI3 2/V 13 = 0 The lines are at a distance 7/V 13 from one another. 5. The equations 0.8x O.4y - 1.2 = 0 and 2x y - 3 = 0 represent the same line; the first equation is obtained from the second by dividing by 512. In the system of equations the two equations are dependent on one another. The lines coincide.







13. Analytic geometry of the pJane


6. The lines), = 2.x - 8 and)' = 2.x 12 are parallel, since ml = 2 = m2 ' Similarly the lines xl4 )'16 = I and x /2 )'/3 = I are parallel, since their Cartesian normaJ forms arc )' = - 3x/2 6 and y = - 3x/2 3. 7. To find the equation of the line that passes through P I {2, - I) and is parallel to the line y = 2.x - 3. - For the required line a point and the gradient m = 2 are given. By using the pointdirection form, one obtains the equation of the line y I = 2{x - 2).






'I1Ie ugIe oflntenection of two IiDeL The angle 'I' = 4:{/l , 12) at which two lines 11 and 12 intersect is obtained most simply from the Hessian normal form; from x cos /PI + y sin /PI - PI = 0 and x cos /Pz + y sin /Pz - pz = 0 it follows immediately that 'I' = 4: {II , Iz) = /Pz - /Pl' From the Cartesian normal form the angle 'I' is obtained, by the restriction to the principaJ value of the inverse tangent function, only to within an additive constant +n. If y = mix + Cl and y = mzx + Cz are the equations of the lines, thenml = tan "'1 andmz = tan "'z , where "'1 = 4: {x, 11), "'z = 4:(x, 12), and so 4:(x, 11) + 4:(/1,/2) = 4:{x, 12), or 4:{ll, 12) = 'I' = "'2 - "'1' By the addition theorem for the tangent function it follows that tan 'I' = tan "'2 - tan "'1 or tan 'P = ml - ml 1 + tan "'1 tan"'2 1 + mlm2


"'1 -

By interchanging the lines one obtains 'P' = 4:(12,/1) = "'2 = -'P or 'P' = n - 'P. This condition includes the condition for the lines to be parallel: from 'P = 0 it follows that ml = m2, which was found earlier. For 'P = n/2 Condition for orthogonaJit ~ - II one ob~s th~ condition for the two lines. to be '-.- -- _ _ _ _ _ _ _y-.l.._m_2_ _ _m _ -l 1. perpendicular. Smce tan 'P = 00, the denommator must be zero, that is, 1 + mlm2 = O. Examples: 1. The lines y - 2 = S(x - 13) and y = - xi S + 18 are perpendicuJar, since in the Cartesian normal form their equations are)' = Sx - 63 and y = - I/ Sx + 18, that is, ml = S is the negative of the reciprocal of ml = - l i S, or ml = - Ilmi' 2. To find the line through the point PI(I, 1) perpendicuJar to the line), = - 2.x/3 + 3. The given line has the gradient ml = - 2/ 3, so the required line has the gradient ml = - l lm, = + 312. The point-

(I _ e 2) Xl

The vertex equation of the circle is also included in this equation. If one puts p = r and e = 0, one obtains y2 = 2rx - x 2 or y2 = x(2r - x); this relation is satisfied, by virtue of the altitude theorem for right-angled triangles. In the common vertex equation a conic is determined by the parameter 2p and the numerical eccentricity e. The quantities used up to now to characterize a conic, the semiaxes a and b and the linear eccentricitye, can be expressed in terms ofp and e if one considers that Yo = 0 gives Xo = 2a and that p = b2 /a for the ellipse and hyperbola and p = r for the circle. One finds for the ellipse a


p/(l - e 2), b


p/V(l - e 2), e


pe/(l _


and for the hyperbola a = p/(e 2 - I), b = p/V(e 2 - I), e = pe/(e 2 - 1); if one chooses p = I, then, for e = 0.8, for example, the rounded-off values are a = 2.78, b = 1.67, e = 2.22, while for e = l.S, a = 0.8, b = 0.89, e = 1.2 (Fig.).

13.5-23 Dependence of a conic on the numerical eccentricity


13. Analytic geometry of the plane Polar equations of the conics

To describe the conic in polar coordinates it is natural to take its axis as the zero direction ; for the ellipse and hyperbola one could choose the centre as pole, but it is more usual to take a focus as pole. Polar equations of conics referred to the centre as pole. The central equation of the ellipse x 2 /a 2 y2/b 2 = 1 is transformed to polar coordinates by putting x = r cos q;, y = r sin q;, where the pole is the centre of the ellipse, so the polar equation is: (r 2/0 2) cos 2 q; + (r2/b 2) sin 2 q; = 1 or 1 = (r2/b 2) (b 2 cos 2 q; + a2 sin 2 q;)/a 2 = (r2/b 2) (b 2 cos 2 q; + a2 - 0 2 cos 2 q;)/a 2 = (r2/b 2) [a 2 - (a 2 - b 2) cos 2 q;)/a 2 = (r2/b 2) [1 - (e 2/0 2) cos 2 q;) = (r2/b 2) (1 - £2 cos 2 q;) , that is, r2 = b 2/(1 - £2 cos 2 q;). Polar equation, The polar equation of the hyreferred to the centre perbola can be obtained similarly.


Polar equations ofthe conics referred to a focus as pole. These conic equations find many applications in astronomy, particularly because of Kepler's first law, which states that the planets move in ellipses having the sun as one focus. One naturally uses as coordinates of planetary motion the distance from the sun and the angle in the orbit, and so one uses a polar coordinate system whose pole is one focus of the ellipse. At the same time, the numerical eccentricity £ is used in astronomy as a measure of the deviation of the elliptic orbit from a circular path. The word eccentricity is a happy choice : in a circle, the centre coincides with the centre of gravitation; the longer the ellipse is stretched, the further is the centre from the centre of gravitation, and so the more eccentric is the path. KEPLER discovered the fact that the planets actually move in ellipses, not circles, by considering Mars which, of all the planets then known, has the greatest eccentricity, £ = 0.0933. The eccentricity of the orbit of the Earth is only £ = 0.0168. Also meteors, comets and artificial satellites, if they have periodic motion inside the solar system, move in elliptical orbits. If they are not periodic, that is, their kinetic energy is sufficient to take them outside the solar system, then they move in parabolas or hyperbolas, provided that one neglects the disturbance caused by the force of attraction of the planets. Polar equation of the ellipse. In Fig. 13.5-8 the focus Fl is taken as the pole of a polar coordinate system, whose zero direction is that of the x,axis from Fl to VI' In the triangle F1 PF2 , since r2 = 20 - rl and IF2Fli = 2e, the cosine law gives : (20 - rl)2 = (2e)2 + ri + 2· 2erl cosq:> or 40 2 - 4arl + ri = 4e 2 + ri + 4erl cosq:> , rl = (a 2 - e 2)/(a + e cos q;) = 0(1 - £2)/(1 + £ cos q;) = b 2/[a(1 + £ cos q;») = p/(1 + £ cos q:» , on putting £ = e/o and b 2 /a = p . Polar equation of the hyperbola. In Fig. 13.5-13 the focus Fl is taken as the pole of a polar coordinate system, whose zero direction is that of the -x-axis from Fl to VI ' In the triangle F1 PF2 , since r2 = 2a + rl and IF2Fli = 2e, the cosine law gives: (2a + rl)2 = (2e)2 + ri - 2· 2erl cos q:> or 40 2 + 40rl + ri = 4e 2 + ri - 4erl cos q:> , rl = (e 2 - a2)/(0 + e cos q;) = b 2/[a(1 + £ cos q:») = p/(1 + £cos q:». Polar equation of the parabola. In Fig. 13.5-4 the focus F is taken as the pole of a polar coordinate system whose zero direction is that of the -x-axis from F to V. Since ILoF I = p, the definition of the parabola gives p-rcosq;=r or r=p/(1+cosq;). All the conics therefore have equations of the same form r = p/(1 + £ cos q;) in a polar coordinate system whose zero direction goes from the pole to the nearest vertex; they differ in the values of the numerical eccentricity, which for an ellipse is positive but less than 1, for a hyperbola is greater than 1 and for a parabola is equal to 1. Also, the circle can be included by taking £ = 0, so that the radius vector has the constant value r = p. Polar equation of the conics referred to a focus as pole



p/( I

+ £ cos q:»

> I hyperbola = I parabola o< < I ellipse

£ £


e = 0 circle

13.5. The conics


Example: The perihelion is defiiied as the point of a planetary orbit nearest to the sun, and the aphelion the furthest point. What is the distance of the aphelion of Mars from the sun1- From

astronomicaJ observations it is known that the major semi-axis a of the orbit of Mars is, in round figures, 1..52 radii of the Earth's orbit (I radius of the Earth's orbit is about 92.6 million miles) and its ec:centricity e = 0.0933. At the aphelion q; = 11. Since p = b1/a = a(b 2 /a2) = a(a2 - ( 2 )/a2 = a(l - e2 ), r = a(1 - ( 2 )/(1 - e) = a(1 + e) = 1.S2 X 1.0933 All 1.66, measured in radii of the Earth's orbit . This means that the distance of Mars from the sun at aphelion is about tS4 . 106 miles. The eccentric anomaly. In astronomy and in the calculation of the elliptic paths or artificial satellites, the eccentric anomaly E, introduced by KEPLER , is used. This is the angle E measured from the zero direction to CP", where C is the centre of the ellipse and P" is the point of the auxiliary circle that corresponds to a point P of the ellipse (Fig.). In plane geometry the construction of an ellipse is carried out from the auxiliary circle with radius a (half the major axis) and the concentric circle with radius b = V(a 2 - e 2 ) (half the minor axis). As was shown in the parametric representation of the ellipse, all its chords perpendicular to the major axis V2 VI are in the ratio b : a to the corresponding chords of the auxiliary circle. If P is a point of the ellipse, the segments IP'PI and IP'p"l are half these chords, I 'P"I = b: a. In the right-angled triangle shown in so IP'PI : P the figure, IP'PI = r sin rp, IP'P"I = a sin E, and so ba sin E = ar sin rp or r sin rp = b sin E. On the major axis, because ICFII = e and ICP'I = a cos E, one obtains r cos rp = a cos E - e. By using the equations 1 = sin 2 rp + cos 2 rp and e 2 = a2 - b 2 , r can be expressed as a function of E : r2 = b 2 sin 2 E + (a 2 cos 2 E - 2ae cos E + e 2 ) = b 2 sin 2 E + a 2 cos 2 E - 2ae cos E + a 2 - b 2 sin 2 E - b 2 cos 2 E = (a2 -. b 2 ) cos 2 E - 2ae cos E + a 2 = (a _ e cos E)2, 13.5-24 Eccentric anomaly and since a e and r 0, r = a - e cos E. This equation contains Kepler's first law, according to which the planets move round the sun in elliptic orbits with the sun as one focus. The relation between the anomaly rp and the eccentric anomaly E is given by the two equations



cos rp = (1/r) (a cos E - e) = (a cos E - e)/(a - e cos E), sin rp = (l/r) b sin E = V(a 2 - e 2) sin E/(a - e cos E), which can also be expressed in the form tan (rp/2) = V[(a + e)/(a - e)) tan (E/2). To obtain the time t as a function of E, one of these equations, the second, for example, is differentiated with respect to t, where, as usual, differentiation with respect to t is denoted by a dot: . b cos E· E(a - e cos E) - e sin E· Eb sin E cos rp . 'P = (a _ e cos E)2

= b . E. It follows that ¢


= dt =

a cos E - e cos 2 E - e sin 2 E = b . E. a cos E - e (a - e cos E)2 (a - e cos E)2

E a cos E - e a - e cos E b . (a _ e cos E)2 . a cos E - e or ¢


= dt =

bE a - e cos E



By Kepler's second law, the area covered by the radius vector in a given time is constant : r2¢ = C. Introducing C into the last relation gives dE r2¢ C C b E= dt = br = br = b(a _ ecosE) or d/= C(a - ecos E) dE, and so the required function 1 = I(E) is obtained by inlegralion: 1=



(Ea - e sin E)


V(a 2




(Ea - e sin E) .

13. Analytic geometry of the plane


As E increases from 0 to 2n, the orbital time T is obtained: T= .!!..-. "-a = 2na V(a 2 - e 2) C "'" C . By Kepler's third law, for each planet there exists a constant p./(41l 2) for which a 2/T2 = p./(41l 2), or, by substituting the above value for T, p. = aC 2/(a 2 - e 2). Therefore three of the four constants are sufficient for all the relations; it is usual to choose e = e/a, C and p. and to derive from r = p/(I + e cos 91) = b 2/[a(l + ecos 91)] : r = C 2/[p.(I + ecos!p)] = C 2(1 - ecosE)/[p.(I - e2 )], cos 91 = (-e + cos E)/(I - e cos E), sin 91 = V(I - e2) sin E/(I - ecos E), t = C 3(E - e sin E)/[p.2(1 - e 2)3/2]. Discussion of the general equation of the second degree The general equation of the second degree in two variables x and y has the form ax 2 + 2bxy + cy2 + 2dx + 2ey + /= 0, where a, b, c, d, e,/are arbitrary real coefficients. It is actual1y of the second degree only when a, b, c are not all zero. This equation defines a curve in the x, y-coordinate system. Henceforth a rectangular coordinate system will always be assumed. The type of curve depends on the values of the coefficients. The discussion, by which one means the characterization of the curve depending on the coefficients, shows the validity of the following theorem. The general equation 0/ the second degree always represents a conic. Elimination of the mixed term. By a rotation of the coordinate system, that is, by a transformation x = .; cos

= 0,

A = O,C + O:

there are three possibilities.

Case I: D", O. Then c and d can be chosen so that Cd + E = 0, Cd 2 + 2Dc + 2Ed + F = O. The equation becomes "12 = -2(D/C)';, and the curve is a parabola.

13.S. The conics


Case 2: D = O. The equation is a quadratic in T), and it therefore represents a pair of parallel lines. They coincide, and therefore represent a double line if A

+ 0, C = 0 : = 0,



0 : Case 1: D and E not both zero. The curve is a single line. Case 2 : D = E = O. Then F must also be zero. Note: The case of a pair of parallel lines can be regarded as a special case of a section of a cone by a .plane parallel to the axis, where the vertex of the cone is at infinity, and the cone is therefore a cylinder. A



Case 1 : E + O. The curve is a parabola. Case 2: E = o. The curve is a pair of parallel lines or a double line.

Discussion of the equation Ax' AC+ 0


N = O

N< O

A = 0, C+ 0


ellip e

A > 0, C>O

N> O

AC = O

+ Cy2 + 2Dx + 2Ey + F =

D2 E2 = 7 + c - F

+ 0, C =


A = 0, C = 0

A < 0, C< 0

no reaJ curve

AC < O


AC > O


AC < O

pair of intersecting lines

A > 0, C > O

no real curve

A < 0, C < 0


AC < O




D = O

pair of parallel lines, coinciding if E2 - FC = 0

E+ O


E= O

pair of parallel lines, coinciding if D2 - FA = 0

D and E not both zero


D= E= O




Example J: In the equation 3x 2 - 30x 8y 65 = 0, the values of the coefficients are A = 3 + 0, C = 0, D 0; the curve is therefore a parabola. To find the vertex, focus and parameter, one divides by 3 and completes the square; from the equation Xl - lOx 25 = -(8/ 3) y - 65/3 75/ 3 or (x - 5)2 = - (8/ 3) (y - S/4) one sees that the parabola is concave downwards, that the vertex is at V(5, S/4) and that the parameter is p = 4/ 3. Example 2: The equation 25x 2 49y2 I SOx - 196y - 804 = 0 describes an ellipse, whose O. The equation can principal axis is parallel to the x-axis, since A = 25 + 0, C = 49 0, N be brought to the central form by twice completing the square: (x + 3)2/49 (y - 2)2/2S = 1. The centre of the ellipse is at C(- 3, 2). The major semi-axis is a = 7, and the minor semi-axis is b = S. 2S6x JOOy - 2244 = 0 represents a hyperbola, Example 3: The equation 64x 2 - 25yl as one sees inunediately. From the equation it follows that 64(x 2 4x) - 2S(y2 - 12y) = 2244. 4) - 2S(yl - 12y 36) = 2244 By completing the square twice, one finds that 64(x 2 4x + 256 - 900 or 64(x + 2)2 - 2S(y - 6)1 = 1600. The central equation is therefore (x 2)2/2S - (y - 6)2/64 = 1. The major axis is parallel to the x·axis, the centre is at C(- 2, 6) and the semi-axes are Sand 8. Example 4: For the conic 9Xl - 4yl = 0, AC + 0 but N = O. Since AC 0, the conic is a pair of interseC/illg lines. In fact, 9Xl - 4yl = (3x - 2y) (3x 2y) = O. Each factor gives a line, and the equations of the lines arey = (3/2) x andy = - (3/2)x. The two lines intersect at the origin.

















Subsequences. If PI, P2 , P3, . .. , P., .. . is any strictly monotonic increasing infinite sequence of natural numbers, then {P.} is called a subsequence of the sequence of natural numbers; for example, the sequence I, 3, 7, 9, 13, 14,27, ... If such a sequence {Po} of indices is chosen, this determines from any sequence {an} one of its subsequences {apo} ' For example, I, 1/8, 1/64, .. . is a subsequence of the sequence I, 1/2, 1/4, 1/8, 1/16, .. . If the terms a" for all n N(e) lie in the e-neighbourhood of the limit a, so that lao - al < e, then the terms aPn of the subsequence with P. > N(e) also lie in this neighbourhood. Hence the following theorem holds.



subsequence {apn} 01 a convergent sequence {a. } -+ a converges 10 the same limit a.

Theorems about convergent sequences. The convergence of the sequence {an} is decided by the existence of a subscript N(e) beyond which lao - al e; the size of N(e) is completely immaterial.

N I , and similarly, for the sequence {b.} , an index N2 so that mined so that la. - al b. - bl e/2 if n N 2 . For all n (NI ,N2), it follows from the triangle inequality that l(a.+b.)-(a+b)1 = I(a. - a) + (b. - max b)IO;;;; la. - al + lb. - bl e, as required. To show that la./b. - a/bl can be made smaller than any given positive number e, one first notes that la./b. - a/bl = I[b(a. - a) - a(b. - b)]f(b ' b.)l o;;;; [lbl·la. - al + lal·lb. - bl]/(lbl · lb.I). N3 can be determined so that Ib.1 ;;;. g 0 for all n N 3 ; this is always possible since b =F O. Finally NI can be determined so that la. - al ge/2 for all n N I , and N2 so that lb. - bl


18. Sequences, series, limits

< glbl e/(2Ial> for all n


max (NI , N




1. If C, CI ' and Cl are constants, and {a.} -+ a, {b.} -+ b, then {ca.} co and {c ia. c2b. } ci a c1b. 2. Since the sequence of the products o f the terms o f two convergent sequences converges to the product of their limits, il f ollows that {a!} tt f or every positive integer k w henever {a"} -+ a . If a" =1= 0 and a =1= 0, this also h olds f or ellery negalive integer k . One can ellen d educe that {a:} -+ a" for every real number if a" =1= 0, a =1= O. J . If {a.} and {b. } are null sequences, so are the sequences {a. b.l, {a. - b.} and {a.b.}. -+



The sequence {a./b.} formed from the null sequences {a.} and {b"} is not, in general, a null sequence. For example, {a.} = {1/2·} and {b.} = {1/4'} are null sequences, but {a./b.} = {2'} is definitely divergent. H the sequences (a~) and (o~') converge to the same limit 0, and the relation 0; ~ 0" ~ 0;' bolds for almost aU terms of tbe sequel!« (II.). then {II.} also converges to the limit II. Corresponding to an arbitrary e 0 there exists an N(e) beyond which all terms of the sequence {a;} and all terms of the sequence {a~'} lie in the e-neighbourhood of a. Since a; .:;:;; a.':;:;; a~', almost all terms of the sequence {a.} also lie in this neighbourhood, so that lim a. = a •


• -+00

Limits of some important convergent sequences

lim;q = I, for arbitrary q ' - + 00


IimVn = I • -+ 00

" - I} is a null sequence. For q = 1 every term 1. For arbitrary positive values of q, {x.} = {)'q

> > <
2"' and m> 2C s. (1 + 1/2) + (1 /3 1/4) (I /S 1/8) 1/2"') 1/ 2 2 . (1 /4) + 4· (1 /8) + ... 210 - 1 (1 /2'") = m/2; s. C.





+ ... +


+ ... + ...( +



Example 2: The series E I/ [(n - 1) n) = 1/(1 . 2) + 1/(2· 3) + 1/(3 . 4) + ... has the nth n- 2 partial sum s. = 1/(1 . 2) + 1/(2' 3) + ... + I/ [n(n + I») = (I - 1/2) + (1 / 2 - 1/ 3) + (1 / 3 - 1/4) + ... + [I /n - I/(n + I)) = I - I/(n + 1). Since {l/(n + I» is a null sequence, the sequence {s. ) = II - I/(n + I» is bounded. The given series converges and has Ihe sum 1. 00


18. Sequences, series, limits

Comparison test. A series whose terms are not smaller than those of a given series with positive terms is said to dominate or majorize the given series. If it converges, then by the first test for convergence its sequence of partial sums is bounded. The fact that it dominates the given series implies that the sequence of partial sums of that series is also bounded, and thus also converges. In exactly the same way one can conclude that a given series with positive terms diverges if there is a corresponding divergent comparison series whose terms are not greater than those of the given series. Such a comparison series is said to be subordinate to the given series.

It Is sufBdent for abe converaeace of a aeries that It Is dominated by a converpot aeries; It Is sufBdent for tile divergence of a aeries that It domJnates a divergent series. 00

1: l In,



E"" Iflln


1/ [n(n

diverges for

,;;;; I






converges for



+ J)I,


In order to be able to use comparison tests, one must have available a sufficiently large supply of ex>

known convergent or divergent series. Series of the form E lIn'" can often be used as comparison 00 ,.-11 series. The series 1: IIn 2 converges, since IIn 2 = I/(n ' n) 1/[(n - I) nl, and it is therefore 11-1



with arbitrary terms converges If It satisfies one of the following CODClItions:

111"+./11,,1~ q < 1

for aU " ;;;. " 0; Um


111,,+.111,,1< 1; ylll,,1~ q
a2, ... , aft all occur in the first N rows of the array, lall

+ la21 + ... + laftl,,;:;;; C1 + C1 + ... + CN'

Because E Ct is assumed to be convergent, the right-hand side is bounded, and it follows that the nth partial sum of the series E laftl on the left-hand side is also bounded, and consequently E aft is absolutely convergent. The 'column series'





Eat, = s" as partial series of E aft> are also absolutely 00

convergent, and Is,l = I E at,l";:;;; E lat,l · From this it follows that the nth partial sum of E s, cer1-1



tainly does not exceed the sum of the series E laftl; this means, however, that E s, is absolutely convergent. Finally, the series E s, and E Zt have the same sum, since each sum is equal to the sum of E aft. Because of the absolute convergence of E aft, according to the second main test one can choose the index m so that for all k;;;;. I, la"'+11 + Ia.. +2 I + ... + la"'+tl e. One now determines N so that the terms ai, a2 , .. . , a", all occur in the first N rows of the above array. Denoting the nth

m occur


If Zt -

uftl is less than e for all n;;;;' N, since



in this expression. Thus, lim E Zt IJ-+ook-l


calculation for the column series yields lim E s, = lim Uft = s. "_00



lim Uft ,.~oo

= s.

An analogous


Multiplication of series. If one multiplies every term of the series 00


E a, by every term of the series


E b" one obtains the partial products indicated in the array below. Each row of the array contains


infinitely many terms all having the same a, as a factor, and each column contains infinitely many terms all having the same bJ as a factor. The product of the two series is now defined to be the series




where Ct is tht: sum of the partial products in the kth diagonal of the array as indicated. For

example, CI = al bl ,c2 = al b2 + a2bl ,C3 = al b3 + a2b2 + a3bl, ... ,Ct = E a,b» ... These partial I+/-k+l products can be found by a translation method, in which one series is written in reverse order and the other is written on a strip of paper which is moved along above the first one. The diagram shows the position for the third term of two product series C3 = a l b3 + a2b2 + a3bl'


If the seriu E a,

'- 1 E C. = C, with c. =

=A co


and E bJ



k - I

1+/- k+1

1- 1

= B are

both absolutely co_rgent, then the product series

a,b) , is also absolutely conwrgent and has the sum C

= AB.

The following example shows that the convergence of the two 'factor series' is not enough to ensure the convergence of the product series. Example: The square of the convergent, but not absolutely convergent, series 1 - 1/'12 + I /V3 - 1/'14 + ... is divergent, because its terms do not form a null sequence. The general term e. of the product of the series with itself satisfies


+ [1 /V3) . [I /V(n - 2») + ... + [I /Vn) • 1 + ... + [l IVn) • [ I /'In) = I.

le.1 = 1 • [I /Vn) [1 /V2) • [I /V(n - 1») ;;. [I /Vn) . [I /Vn) [I /Vn) • [I /'In]


18.3. Limit of a function-Continuity Limit and continuity of a function are concepts without which a rigorous construction of higber analysis is impossible. If a function describes a physical situation, then the concepts of limit and continuity often have a physical meaning also.

18.3. Limit of a function-Continuity


Umit of a function Limit at a point. The concept of the limit of a function y = I(x) can be related to the concept of the limit of a sequence. This is done by allowing the independent variable x to run through a convergent sequence of numbers {x.} tending to the limit a (the abscissa sequence), and considering the ordinate sequence {/(x.)} of the values I(x,) of the function corresponding to the x, . If the convergence behaviour of the ordinate sequence {/(x.)} depends upon the choice of abscissa sequence, that is, if two different abscissa sequences both converging to a have corresponding ordinate sequences converging to different limits, or if an ordinate sequence diverges, then the function I(x) does not tend to a limit as x tends to a. On the other hand, if the ordinate sequence {/(x.)} tends to L for every abscissa sequence {x.} tending to a, one says that the function I(x) has the limit L as x -. a. This means that the values I(x) of the function come the nearer to the number L the nearer the argument x comes to the value a. The difference I/(x) - LI bey tween the value of the function and the limit is less than every arbitrarily chosen positive number e, provided that the value of x differs from a by less than a suitably chosen number 6 = O(e) depending on e, that is, provided that 0 < Ix - al L ~£ 6(e) (Fig.). The number 6(e) is by no means uniquely - -L-'-*- -"""'Ioo;;;:::c- - - - - r - - determined, for if one 6(e) with the required properties has L-£ been found, then clearly every smaller number 6' 6(e) will also serve.

0, such that the inequality I/(x) - L I < 11 bolds for every x satisfying the condition 0 < 1.1: - "l < 6(e).






18.3-1 Geometrical illustration of the limit concept

Exa'!1'le 1: The function Xl has the limit zero as the argument x tends to zero, lim x 2 = 0, since Ix - 01 e for aU x such that Ix - 01< O(B) '" VB. x_O Example 2: The function l /x tends to the limit l /a as x tends to 0 ~ 0, lim I/x = J/o. This


}if to every positive number

(or f(x)

< - )for all x

> 0),

,however large, there corresponds

with 0

< Ix -


< o(


Example: The tangent fun ction y = tan x is not defined for x = n/2. but has both a right-hand and a left-hand infinite limit for this value ; lim ta n x = + 00, lim tan x = : I/t. This example shows that the concept of the limit L of a function f(x) can be extended to the case of unbounded increasing (or decreasing) abscissae. 0 there corresponds a sufficiently large wee} 0 such lim f(x} = L if to every arbitrary E

> > > wee). Similarly. x_lim- f(x} = L if to every arbitrary E > 0 there a sufficielllly large wee) > 0 such that If(x) - LI < t for all x < - Wet}.

rhat If (x) - LI corresponds

Example 2: The limit lim in x does not exist. No mailer how large a value of x, say Xo, is x_ ex>

chosen, beca use of the periodicity of the si ne function there are always infinitely many abscissae greater than Xo for which the function takes any prescribed value between - I and + 1. The behaviour of rational functions at infinity is dealt with in Chapter 5. Calculations with limits. The rules drawn up for calculation with limits of sequences can be carried over word-for-word to calculations with limits of functions. These rules, which have already been applied in the examples of the previous section, state that the operation of forming the limit can be interchanged with addition, subtraction, multiplication and division (if L 9= 0), provided that all the limits occurring exist and are finite. The first two rules hold also for sums and products of several functions, but not necessarily for infinite sums. A function hex) whose values in a neighbourhood of the point a lie between those of two functions f(x) and g(x) that both have the limit L as x ...... a also has the limit L. Iimf(x) = F and lim g(x) = G

lim ([(x)


x _a


lim ([(x) . g(x)] = limf(x) . lim g(x) = F· G, x-a

.. _ 0


± g(x» ) =

lim f(x)


± lim g(x) = F ± G, x_o

limf(x}/g(x) = limf(x)/Iim g(x) = FIG if G =!= 0




= L and lim g(x) = L. and if the inequality f(x} ~ hex) ~ g(x) holds in a neighbourX_G hood of a, then lim hex) = L also. If limf(x} X~G

Example J: lim (sin x)/x = O. Because sin x lies



between - 1 and + 1, for x 0 holds the inequality - l lx ~ (sin x}/x ~ Ilx. The result follows, since lim 11x = lim (- I/x) = 0 (Fig.).

--- - -- 3rr


Example 2: lim x sin l/x = O, since -Ix l ~ x sin l /x .. _ 0

~ Ixl and lim Ix l = lim (- Ix l> = %_0

.. .... 0



sin x

18.3-4 Graph of the function y - -x-

18.3. Limit of a function-Continuity


Some important limits For the determination of the limit of a function there are hardly any generally applicable methods. Some frequently used limits will be derived here. with the help of knowledge of convergent number sequences. 1.


1 aX -+ 1 as x -+ 0 if a > o. I

!i:o aX = I

for a



It has already been shown that the sequence {Va} for a> 0 converges to 1. The sequence {I/Va} then has the reciprocal limit. likewise equal to 1. It follows that to every arbitrary e 0 a positive integer N can be determined. so that for all n;;;' N. the numbers all' and a-II' lie in the interval I/N from 1 - e to 1 + e. Since the exponential function is monotonic. all aX with -1/N x also lie in this interval. Thus. 1 - e aX 1 + e. or laX - 11 Ii if Ixl o(e) = I/N.


2. 1 (1

+ I/x)X -+ e

o. such that the inequality satisfied for all x with Ix - al O(e). Firstly. since b> 1. for positive Ii 1 and ez = 1 - b- e are also positive. With b" 1 it follows further that b-') = el' Now let e 0 be arbitrary. Then one can choose o(e) = aez

2/(n6») with the following property : for nl = 2k, n2 = 4k + I, n, = 4k + 3 (k an integer) the function sin (I/x) = sin nn/2 takes the values 0, +1, -I (Fig.). The function oscillates between +1 and -1 more and more rapidly, the larger n is or the nearer x approaches to zero. The function


:I I





Y'w" 1

:l I I


: -n ~ :in















18.3-9 Jump from









18.3-10 A discontinuous oscillatory function

18.3-11 An oscillatory function with removable discontinuity

therefore has no limit as x -+ 0; the discontinuity at that point is not removable. On the other hand, the discontinuity of the function x sin (I/xl at the point x = 0 is removable, since lim x sin (1/xl:= o. Consex~o


quendy I*(x) = x sin (l/x) for x =to 0;' 1*(0) = 0 is a continuous replacement function for x sin (l/x) (Fig.). 'Ibeorems about coatiDuous functions. From the rules for calculating with limits the following theorem can be deduced immediately. The sum, difference, ond product 01 two lunctions continuous ot x = t are likewise continuous at this point. Their quotient is continuous provided that the denominator is not zero lor x = t.

Since it is recognized without difficulty that the functions g(x) = c = constant and hex) = x are continuous everywhere, it follows at once from this theorem that all functions obtained from them by means of the four basic operations are continuous. The first two of the following statements about the continuity of the elementary functions are proved in this way. 1. Every polynomiallunction I(x ) = a"x" + a._IX-- 1 + ... + alx + 00 is continuous everywhere. 2 . .A ralionallunction p(x)/q(x) is continuous at oll points t lor which q(t) :+= O. J . The exponentiallunctions I(x) = 0"(0 0) ore continuous everywhere. 4. The logarithmic lunctions I(x) = los. x(b 0; b :+= I) are continuous lor all positive values o/x.



18.3. Limit of a function-Continuity



S. The trigonometric lunctions sin x and cos x are continuous everywhere; the lunction tan x = sin x/cos x is continuous lor all (2k I) n/2 (k an integu) and the lunction hr (k an integer). cot x = cos x/sin x lor all , With the help of the limit lim aX = I already obtained, one deduces that lim aX = lim (a< . aX-

19.1-1 Schematic distance-time diagram for the journey of a train from town A to town B, s distance, t time


Y Yf


,, ,, ,

, -- ---------, Xf-x"




19.1-2 Slope of a curve at the point p.

Definition of the derivative Difference quotient of a function. If a curve in a Cartesian coordinate system is the graph of a function Y = f(x), then each of its points p. (n = 0, 1, 2, ... ) has coordinates x. and y. = f(x.), where the x. belong to the domain of definition of the function (Fig.). One can then form differences such as Lfx = Xl - Xo = hand Lfy = Yl - Yo = f(Xl) - f(xo) = f(xo + Lfx) - f(xo) = f(xo + h) - f(xo), and their quotient Lfy/Lfx has a finite value for XI =1= Xo . It is called a difference quotient, and geometrically it represents the slope tan ,o describes the slope of tl to the +x-axis (Fig.). Similarly z, = :y I(xo, y) = tan 9'x,.>, is the slope with respect to the +y-axis of the tangent t2 to the curve in which the surface z = I(x, y) is cut by the plane x = Xo parallel to the y, z-plane. At each point P of the surface both a tangent tl determined by z" and a tangent t2 determined by Zy are defined. Under certain assumptions, which are usually satisfied in practice, the two tangents span a plane to the surface at the point P. Partial derivatives of higher order. Each partial derivative is again a function of the same variables, and can itself have partial derivatives if the limits of the corresponding difference quotients exist. These are called partial derivatives 01 higher order, for example, of the second, third, .. . nth order. Partial derivatives with respect to different variables are called mixed derivatives. From z = I(x, y), for example, by differentiating aaz = order: x I"

J"" =

a/" ax =


a z ax2'




a/" ay

I" and



aaz = /y one obtains four derivatives of the second y




ai, ax

= ~ and ayax



= a/y = a 2 z ay

ay 2 '

From the way in which they are formed, the functionslxy and/yx are different from one another. But Leonhard EULER (1707-1783) already knew conditions under which they are equal; Hermann Amandus SCHWARZ (1843-1921) proved the theorem named after him. Tbeorem of Schwarz: U the mixed partial derivatives of the second order I", and /y" of a function I(x, y) are continuous functions of x and y In a domain D, I", = I,Jt then they are equal to one another In the interior of this domain.



The continuity of a function u = I(x, y) at the point (xo, Yo) means that corresponding to an arbitrary prescribed e> 0 there always exists a positive number ~ = ~(e) such that /I(x, y) - I(xo, Yo)/';;;; e for all pairs of numbers (x, y) satisfying (x - XO)2 + (y - Yor~ .;;;; ~2 . Geometrically this means that the values of the function I(x, y) differ by an arbitrarily small amount from I(xo, Yo) provided that the argument (x, y) is chosen within a sufficiently small circle with centre at (xo, yo). lf/(x, y) is continuous at every point of a domain D, then thelunction is continuous in D. The theorem of Schwarz holds also for partial derivatives of higher order and also for functions of more than two variables; for example, for z = I(x, y),I""y = Ixy" = I,,,,, andlx " =I,,,y = I yy,,' Example J: z = /{x,y) = Xl 7x2y 3xy' - Sy6, I" = 3x2 + 14xy + 3y 5; 17 = 7x 2 + ISxy4 - 30y'; Izz = 6x + 14y; 1"7 = 14x + lSy4 = /y,,; I" = 6Oxy' - lSOy4; I""" = 6; I"", =1"7" = /7"" = 14; 1"77 =/ 71


e< x









x O

f (n - l l(x)



Y""t I


~ xm

1r ~m




19.4-2 Schematic representation for Points of inHection. The tangent to the curve of the f'(x) = O;f"(x) = 0; f"'(x) '" 0 3 2 function y = (1/6) (x - 3x - 9x + 17) (see Fig. 19.4-1) at the point (-3, -5/3) has the slope [,(-3) = +6. The slope decreases to the value ['(-1) = 0 at the maximum (-1, +11/3), and on passing through this point decreases still further as far as the point Xw = +1, where it has the value /,(1) = -2. From there on the slope increases monotonically. The function [,(x) has a local minimum at Xw = 1. In the interval -00 < x ,;;;; + I the direction of a tangent moves into a direction of a following tangent, in the sense of increasing values of x, by a rotation to the right, in the mathematically negative sense. In the interval + 1 ,;;;; x < +00 there is a corresponding rotation to the left, in the mathematically positive sense. At the point of inflection the sense of rotation changes from right to left. H one imagines driving a vehicle along the curve, then the road lies to the right of the tangent as far as the point of inflection, but after one has passed through this point it lies to the left of the


19. Differential calculus

tangent. The curvature of the curve before the point of inflection is opposite in sign to that after it. If a portion of a curve through three points situated sufficiently close to one another is replaced by a circular are, then the centre of this circle 01 curvature lies to the right of the curve before the point of inflection and to the left of it afterwards. The tangent at the point of inflection or in/lectional tangent thus separates portions of the curve whose curvatures are in opposite senses. From these considerations it follows that the function has a point of inflection where its first derivative assumes an extreme value. If in addition f'(x..,) = 0, then the inflectional tangent is horizontal; one speaks of a horizontal point 01 inflection (Fig.). Tf I "(x w ) = 0 and f'''(x w ) + 0 then x'" is a point of inflection. Equation of the inflectional tangent : (y - y..,) = f'(x ..) (x - xw)' The criteria for extrema can be applied to the first derivative f'(x), regarded as a function IP(x). Consequently a sufficient condition for a local maximum or minimum of f'(x) at the point x.., is 1P/(x..,) = I"(x..,) = 0 and IP"(x..,) = I"/(x..,) < 0 or 1"/(x..,) > O. If1"/(x..,) = 0, then the last theorem above holds, since a derivative of odd order of I(x) is one of even order of IP(x). 19.4-3 Graph of a function with a horizontal point of inflection


x 19.4-4 Points of inflection W,. W, and inflectional tangents of the graph of the function

I .. I.

I(x) = (x' - 2x' - 12x'

+ 8x + 20)/10

The lunclion y = I(x) has a poinl 01 in/leclion at a point'; at which 1"('; ) = 0 il the first non-vanishing derivative I(·)(~) (n 2) is olodd order.


Examp/e 1: The function = 0.I(4x J - 6x 1 - 24x that is, x 2 - X - 2 = 0 one 1 " I(X2) = + 3.6 O. XI = -

= I(x) = 0.1 (x· - 2.x3 - 12.x 2 + 8x + 20) has the derivatives 1 J.2.x - 2.4 and y IN = 2.4x - 1.2. From y" = 0, obtains XI = - I and X2 = + 2. Because 1 "'(xI) = - 3.6 + 0 and I and X2 = + 2 are abscissae of points 01 inflection (Fig.). The y

+ 8), y " = 1.2.x



corresponding ordinates are/(xI) = + 0.3 and/(x2) = - 1.2. The in/lectiona/tangents II and t2 at the points of inflection WI (- I. + 0.3) and W 1 ( + 2. - 1.2) have the slopes !'(x,) = + 2.2 and !'(X2) = - 3.2 and hence have the equations (y - 0.3) = 2.2(" + I), or y = 2.2.x + 2.5 for II and (y + 1.2) = - 3.2(x - 2) or y = - 3.2.x + 5.2 for 12' Example 2: The function y = I(x) = (x 2 - 4)/x has no point of inflection. because I " (x) = 0 is a necessary condition for a point of inflection. but y " = - 81x J cannot have the value zero for any finite value of x. Applications. If one succeeds in expressing a variable I as a continuous and differentiable function of a variable x, then one can calculate for which value of x the variable I has an extreme value. From the given conditions it can be determined whether this value is a local maximum or minimum. In applications, however, it is usually required to find absolute extrema. If the function I(x) is continuous in the closed interval a ~ x ~ b and differentiable in the open interval a x b, then its absolute minimum (or maximum) is either the smallest local minimum, the greatest local maximum or one of the boundary values I(a) or I(b).

0 = VA = al












ExQmple 4: A sector is to be cut out of a circular piece of sheet metal of radius R, and the remainder bent together to form a conical funnel (Fig.). For what angle E at the centre does the funnel have the greatest capacity? The formula V = (n/3) r 2 h for the volume of a cone, together with the additional condition rl = R2 - h 2 , gives the equation V = I(h) = ~n/3) (RIh - h 3 ). For extrema, /,(h,) = 71(R - 3hD/3 = 0; hi = (R13) V3 , r(h) = - mh, r(h l ) = - (2/3) TIR V3 O. Thus, h, gives a maximum. From the addi= (RI 3) V6. In bending tional condition, the sheet the circular arc of length b = eR

O, bl - 4ac =l= 0 =)Ta 2 Va

I . 2aX + b 1 . 2ax + b J = - I/( - a) arcSin V(b2 _ 4ac) + C 1 = "Q arcsln V(4ac _ b2) for a

< 0,

bl - 4ac > 0


a> 0,



bl _ 4ac = O

Substitution x =

x = x =

Examples: 1. The substitution X = sin z, dx = cos z dz, z = arcsin x yields f (1 - x 2 ) dx = f V(I - sin 2 z) cos z dz = f cos1 Z dz = (1 /2) (z + sin z cos z) = (1 /2) (arcsin x x V(1 - x 2 » C.



2. From x = r sin z, dx = r cos z dz one obtains similarly f V(r l - xl) dx = (1 /2) (r2 arcsin x/r x V(r l - Xl»




f f

I/l [ inh- ' x + x (I + Xl)] + C ± (a 1 /2) sinh- Ix /a + (x/ 2) (a l + x 1 ) + C for a :> 0

x x


Ja l _ blxl

(I /ab) tanh- I (bx/a)

C for Ixl


+ C.

Integral 11(1 + x 2) dx (a l + Xl) dx

sin z sinh z cosh z

< la/bl

= sinh z = a sinh z

x = (a/b) tanh z

Integrals of functions R(sin x, cos X, tan x, cot x). Transformation into a rational function of z can be achieved by the substitution z = tan (x/2) : . . 2 tan (x/2) cos 2 (x/2) 2z sm x = 2 sm (x/2) cos (x/2) = sin2(x/2) + cos2 (x/2) = ~ , cos 2 (x/2) - sin 2 (x/2) cos X = cos2 (x/2) + sin2 (x/2) 1 - Z2 cot x = --2-z-'


1 - Z2 1 + Z2 •

tan x =

dz 1 1 + Z2 dx = 2 cos2 (x/2) = --2-

. JR (2Z 1JR(slnx,cosx,tanx,cotx)dx = 1 + Z2' 1 + Z2


2z 2 tan (x/2) 1 - tan 2 (x/2) = 1 - Zl' dx

dz 2z




+ Z2


Zl )

.~ . ~

2 dz 1 + Z2


20. Integral calculus


Examples: 1.





1 - sin x d = sin x(i - cos x) x


I + Z2 2z(1 + Z2) dz =


Z1 _ : :



= In (cz)


In (c· tan x / 2).

+ Z1)] . [2/(1 + Z1)] dz + Z1)] [I - (I - 1 2)/(1 + Z2)] [I /z - 2/Z2 + l /z3] dz = (1 /2) [In Izl + 2/z (I /2) col 2 (xI2)] + c.

[1 - 2z/ (1 [2z/(1


+ 1 dz = (1 / 2) (1 /2) [In Itan (xI2)1 + 2 cot (xI2) -

= (1/2) =

dx sin x = 2



Integrals of functions R(sinh x, cosh x, tanh x, roth x). According to the definitions of the hyperbolic functions, these integrals may be converted into integrals of rational functions by means of the substitution eX = t, for example, sinh x = [t - l/t]/2. By analogy with the trigonometric case the substitution z = tanh (x/2) is also successful: . . 2 tanh (x/2) cosh 2 (x/2) 2z smh x = 2 smh (x/2) cosh (x/2) = cosh2 (x/2) _ sinh2 (x/2) = I _ Z2 ' cosh x = sinh2(x/2) + cosh2 (x/2) = cosh 2 (x/2) - sinh 2 (x/2) 2 tanh (x/2) 2z tanh x


dx =



1 + Z2 1 - Z2 '

+ tanh2 (x/2) = 1 + Z2'

1 2 cosh 2 (x/2)

coth x


1 + Z2

2Z '

cosh 2 (x/2) - sinh 2 (x/2) I - z2 2 cosh 2 (x/2) = --2-'


dz =

2 1 - Z2 .

Binomial integrals f x"'(a + bx")P dx. Here the coefficients a and b are real numbers and the exponents m, nand p are rational numbers. A theorem of P. L. CHEBYSHEV (1821-1894) states that these integrals can be expressed as elementary functions when at least one of the numbers p, (m + I)/n, or (m + 1)/n + p is an integer. If p is an integer, the integrand is a sum of powers with rational exponents which can be integrated. If (m + 1)/n is an integer andp = sir, one puts z = V(a + bx'); if (m

+ 1)/n + p is an integer, one puts z =


+ bx·)/x"].

Integrals that cannot be expressed in terms of elementary functions The calculation of the length of an elliptical are, of the period of oscillation of a circular pendulum, and of other problems lead to elliptic integrals. These are integrals whose integrand contains the square root of a cubic or quartic polynomial with no repeated root.

Joseph LIOUVILLE (1809-1882) proved that they belong to the class of those integrals that cannot be expressed in closed form in terms of elementary functions. There are other integrals of this type I sin x or V(1 + 1 4). But thOIS . h comparatIve . Iy sImp . I· wit e mtegran d s, sue h as cOSiX - cos x x x does not mean that these integrals do not exist; as indefinite integrals of continuous functions they are, as has been shown, differentiable functions of the upper limit of integration. On the contrary, integrals that cannot be expressed by elementary functions are accepted into mathematics as new, higher, non-elementary functions . They are often treated by first expanding the integrand as an infinite series, which is then integrated term by term (see Chapter 21.).


If the infinite series

) , --,


:£ J.(x) converges uniformly on the interval a .;;;: x


.;;;: b and if each term f.(x)

is integrable, then the series obtained by integrating term wise over [a, b] also converges.

20.3. Integration of functions of several variables Since the definite integral is particularly useful for calculating areas of plane regions, it is natural to look for a generalization to facilitate the calculations of volumes of spatial regions. If a bounded continuous function z = f(x., X2, ••• , x.) is defined on a measurable bounded region Gin n-dimen-

20.3. Integration of functions of several variables


sional space. one divides G up into a finite number of measurable subsets and forms. just as in the definition of a simple definite integral. upper and lower sums involving the volumes of these subsets and the maxima and minima of I(x" X2 • ...• x.) in each of the subsets. If these sums approach the same limit as the subdivision is refined. this limit is called the n-Iold volume integral of lover G. The two-fold volume integral. called the double integral. will be discussed in greater detail; it can be used to calculate the volume of solid bodies that are bounded by curved surfaces. However. the range of integration of an n-fold integral can be restricted to a manifold of lower dimension. For example. one speaks of a line integral for n = 3 when this manifold is a I-dimensional curve. or a surlace integral when it is a 2-dimensional surface. Two-dimensional integrals Double integral. The definite integral was defined as a limit of sums in which each term is the product of two factors. the lengths Llx, of subintervals and the ordinates 1(/;,) at a point /;, of the subinterval. the number n of subintervals tending to infinity and the length of the longest subinterval tending to zero. The interval [a. b] of integration on the x-axis is now replaced by a plane region G on which a function z = I(x. y) is defined. and G is divided into n subregions LlG,. i = 1. 2, ...• n. To simplify the notation. one writes LlG, for the subregion and also for its area. Suppose that the function is continuous and bounded in the region G. Then one can form lower sums with the infimum m, in LlG,. and upper sums with the supremum M, in LlG,. If the subdivision of G is refined. then as n --+ 00 and LlG, --+ O. the sequence of lower sums tends to the same limit as II

the sequence of upper sums. and any sequence of intermediate sums}; 1(/;,.1),) LlG, tends to the


same limit whatever intermediate point (/;, .1),) is chosen in LlG,. The integral of the function z = I(x. y) over the region G is defined to be this common limit and is called a double integral. because there are two variables of integration.

II I(x. y) dG = G

Double integral

lim 11_ 00

(x) is a continuous function of x and is therefore integrable over the interval [ai' a2]. Similarly


x, then this limit is called the line integral of the first kind of the funct ionf(x, y . z) along the curve C from A to B. Li ne integral

Jf (x, y , z) d

The calculation of the integral can be reduced to that of a definite integral. If x = x(t). y = Y(/), z = Z(/) is any parametric representation of the curve C (the arc AB corresponding to the parameter interval II ,;;;; I ,;;;; t 2 ) then because :; = V[X(t)2 + Y(/)2




f (x, y, z) ds =


f[x(t ), y(1). Z(I )]


+ i(t)2) one has :



dt =


yet ). z( t )] I [.\ (1)2

+ Y( I )2 + t(I )' ) dt

20.3. Integration of functions of several variables


If P(x, y, Z), Q(X, y, z) and R(x,}" z) are continuous functions, one can define similarly other types of line integral f P(x, y, z) dx, JQ(x, y, z) dy, f R(x, y, z) dz ; for example, the first of these is the


limit of sums





P[X,(S), YI(S), Z,(S)] .dxt. where .dX, is the projection on the x-axis of the ith arc of

subdivision of the curve. If these three integrals are added together, one obtains the line integral of the second kind f[P(x, y, z) dx


+ Q(x, y, z) dy + R(x, y, z) dz).

The calculation of such an integral in two dimensions can sometimes be simplified by applying the following theorem. If P(x,)I) dx + Q(x,)I) d)l Is the total differential dF(x, )I) of a func:ti('n F(x, y), and If P(x, y) and Q(x,)I) are continuous in a connected region G. then the value of the Iino! i.ltegral I [P(x, y) dx


+ Q(x, y) d)l) depends only on the end-points A and B of the path of Integration in G and not on the

particular path Joining A to B. This follows from the fact that f[P(x, y) dx


+ Q(x, y) dy) = f dF(x, y) = f" dF(x, y) = c


FIx(t l ), y(tl») - FIx(t2), y(12»)

depends only on the limits of integration. An equivalent statement is that the line integral f[P dx + Q dy) is zero for every closed curve C in the region G.


The following theorem, which is easy to deduce from Gauss's theorem (see Divergence and theorem of Gauss), provides a criterion for P dx + Q dy to be a total differential. If the region G in question is simply-connected and if the functions P(x,)I) and Q(x, )I) are continuously differentia ble in G, then the integrability condition ficient fo r P(x,)I) dx + Q(x,)I) d)l to be a total differential.

In Chapter 22. it will be shown that if P dx + Q dy is not a total differential, then it is always possible to find an integrating factor p(x, y) such that the product p(x, y)(Pdx + Q dy) is a total differential. Example: To calculate lex dx

+ y dy)

~P Y


~Q x

i necessary and suf-

Integrability condition

along the parabola y =

= 2x dx, and 1 I(x dx + y dy) = J y/> Z,)

" f(x/o y/o Z,) .dS/o where .dS, is the surface area of S" If this sum approaches in St. and forms the sum L' I-I

a limiting value as n ..... 00 and .dS, ..... 0, independent of the choice of the points Pt. this limit is called the surface integral of the function f(x, y, z) over the surface S and is denoted by f f(x, y, z) dS. Surface integral

I f (x, y, z) dS =



J S , -O

"- CO

i f (x /> Y I, Z, ) ,-I



Calculation of a surface integral can be reduced to that of a double integral as follows: one inserts the parametric representation of the coordinates x, y, z into f(x, y, z); the surface element dS (see


20. Integral calculus

Chapter 26.) has the form dS = V(EG - F2), where E = x • • X., F = x• . Xv, G= Therefore l/(x, y, z) dS = 1[X(u, v), y(u, v), z(u, v)] V(EG - F2) du dv.

Xo •



For I = 1, the integral yields the surface area of S. Surface integrals of the second kind are defined analogously to line integrals of the second kind. Applications in mechanics Work. The concept 'work done by a force' is defined in terms of a line integral of the second kind. The force F is a special vector field (see Vector analysis); let F", F" Fa be its components referred to a Cartesian (x, y, z)-coordinate system. If P(x, y, z), the point at which the force F is applied, moves along a smooth path C, then W = l(F" dx + F, dy + F. dz) is called the work done by the

force F along the path C (Fig.). If the vector with components dx, dy, dz is denoted by dr, then the integral for work done can be written vectorially, using the scalar product F' dr. Work done is the line il/tegral 01 the lorce.


Work as integral

F . dr



20.3-14 Indicator diagram of a steam engine

For instance, if the force F is constant and the path C is a segment starting at the origin and represented by a vector r of length Irl, then the angle rp between the directions of F and r is also constant and the work integral leads to the formula W = IF I . Irl . cos rp; in this case the work done is the product of the component of force IFI cos rp along the direction of the path and the path length Irl . In physical problems the components of force are usually the partial derivatives of a function V called the potential. Work done is then the line integral of a total differential and depends only on the end-points of the path. The field of force is then called conservative. Example J: What is the work done in extending a spiral spring by I units of length if the force acts in the direction of the spring? - Denoting the spring constant by D, the force is F= Dx and

the work done W =

J Fdx = D i x dx = DI 2 /2. I

Example 2: To accelerate a body of mass m from the velocity

work W ; since F = m




W =

Fds =



to the velocity



• one obtains

J ".




Jvdv = (v~ -vn·m/2. p.

ds = m


The work done is equal to the increase in kinetic energy. Static moment. The static moment M of a point mass about an axis is defined to be the product of the distance I of the point mass from the axis and of the mass m. Static moment of a continuous mass distribution M =


20.3-15 Static moment of a region

J I dm = (! JI d V

For the static moment dM of an element of mass dm of a continuous mass distribution, the differential expression dM = I· dm holds, and the static mOlJlent of the entire mass is obtained by integration. If (! is the constant density and d V the volume element, then dm = (! d V. To define the static moments about each of the coordinate axes for the region below the curve y = I(x), one simply calculates the moments for a continuous mass distribution of density (! = 1 over a body

20.3. Integration of functions of several variables


of thickness d = 1 lying above the region (Fig.). To calculate the moment about the y-axis of the region under the curve y = f(x), one decomposes the region into strips of width Lix = dx. The area of the strip, by the mean value theorem of the integral calculus, isf(~) dx for some intermediate value ~ in Lix, and the moment of the strip is therefore dM = U(~) dx. Integration then gives the moment of the region. To obtain the moment about the x-axis, one decomposes each of the above strips into elements of breadth Liy = dy. This element has the moment dM = 'YJ dy dx, where 'YJ is an intermediate value in Liy, and integration in the y-direction yields dM

= 1'YJ dy dx

for the

moment of the strip about the x-axis. Finally, integration over x yields the total moment of the region. Static moment about the y-axis of the region below the curve

y = f(x) between a and b

Static moment about the x-axis of the region beneath the curve Y = f(x) between a and b



= J xydx a

b )'




I I '1 dydx = +I y 1 dx a 0


Slmtlarly the statIc moment of a curve can be obtamed by consldermg a umform mass dlstnbutlon of density e = 1 along the curve; the moments about the x-axis and the y-axis are Mx =


f y V(I + y'2) dx


and My =


f x V(I + y '2) dx,


respectively. For a body of revolution about the x-axis, one can calculate the static moment with respect to the plane through the origin perpendicular to the x-axis as b

M = n f xy 2 dx.


Centre of mass. Every solid body can be regarded as a system of point masses, and there is always one point, the centre of mass, at which one can imagine the entire mass of the body to be concentrated. The static moment of a continuous mass distribution is equal to the static moment of the centre of mass with respect to the same axis: M = f I dm = Ic m. By using the appropriate static moments, m

one can obtain the coordinates of the centre of mass for uniform distributions along a plane curve, over a plane region, and a solid of revolution : Xc = My/s and Yc = M,js ; Xc = My/A and Yc = Mx/A; Xc = M/V. Coordinates (xc. Y c) of the centre of mass a) uniform distribu tion on a plane curve b

b) uniform distribution below the curve y = f (x)

+ y' Z) dx ; + y'2) dx

M J x V( I x = --y = " c S V( I




Xc =



= _a_b- _ ;

f ydx


f V(I + y'Z) dx

xc = v


f xyl dx =


f y V(I + y'Z) dx



f xydx


Yc = T


f y2 dx




2fa y dx



; 1





M Yc = --" =

c) uniform solid of revolution about the x-axis






For the solid of revolution about the x-axis, the centre of mass lies on the x-axis, that is, Yc = Zc = O. Example: To calculate the coordinate of the centre of mass of the region below the curve y = f(x) = cos x between 0 and n/2. - The area is A =

be found after an integration by parts in each case:

1~os x dx =

J. The required integrals can

i x cos x dx = n/2 ,,/1



Lcosz x dx =


The formulae for the coordinates of the centre of mass then lead to Xc = n/2 - 1, Yc = n/ 8. The formulae lead to Pappus' rules. The generating region for the solid of revolution about the x-axis has the static moment M"


= (1/2>J yl dx and the ordinate of the centre of mass is Yc = II



20. Integral calculus

This gives, for the volume of the solid of revolution, the relation V.. = n


f yl dx =


2n:. III


f yl dx =



= 2n:YcA .

Pappus' rule for the volume of a solid of revolution: the volume is the product of the area revolved and the length of the path described by its centre of mass. The surface of the solid of revolution is generated by revolving the curve y = f(x) about the

x-axis; this curve has the static moment M ..


= f y V(I + y'l ) dx

and its centre of mass has the

coordinate Yc = M It follows that the surface area S.. is given by S .. = 2n: f y V(I + y'l) dx = 2n:M.. = 2n:ycS. " Pappus' rule for the area of a surface of revolution: the surfoce area is the product of the lengt" of the generating curlle and the length of the path described by its cel/tre of mass. Moment of inertia. The kinetic energy W of a body of mass M and velocity v is W = v2 M 12. If a rigid body rotates abou t a fixed axis A, its vario us portio ns have different velocities. If w denotes the constant angular velocity and x the distance of the element of mass dm fro m the ax is of rotation , then this element has the velocity v = x w and the kinetic energy d W = (1 / 2) X 2W 2 dm. The ki netic energy of the whole body is obtained by integrat ion as W = ( 1/2) w 2 f x 2 d m, where the integration is taken over all elements of mass. m "

axial Moment of inertia polar

IA= fx 2 dm m

Ip= Jr 2 dm nt



dm rna element; x or r di tance from axi A of rotation or point P of reference


20.3- 16 Momen t of inert ia


If one compares the two expressions for kinetic energy. one notices that the mass M has been replaced by the integral J x 2 dm; this is calleCl the axial moment oj inertia JA with respect to the axis ot rotation A (Fig.). If the moment of inertia is defined not with respect to a reference axis A but with respect to a reference point P, one obtains the polar moment of inertia I p . An important relation between the polar moment of inertia Ip with respect to the origin 0 of a rectangular Cartesian coordinate system and the axial moments of inertia I .. and I. with respect to the two coordinate axes can be obtained by using the relation rl = x 2 y2, where r, x, yare

the distances of a mass element from the origin and from the axes. Thus, Ip = J r 2 dm = f (Xl + yl) dm = f xl dm + f y2 dm = I .. + Iym






= I" + I .

Example I : To find the moment of inertia of a thin straight rod of length I, cro -sectional area q and uniform density (!. with respect to an axis passing through one end of the rod and at right angles to it (Fig. 20.3-16). [f dx is an element of length of the rod. the corresponding element

of mass is dm

= (!q dx.

Hence IA =

JX l



dm = Jxl(!q dm = (!q

rx2 dm = qf.!./l/ 3 = M1 2/ 3. 0 I

since the total mass of the rod is M = eql. Example 2: To find the moments of inertia of a thin circular plate of diameter d with respect to the centre and with respect to one of the li nes through the centre (Fig. 20.3-16). To simplify matters. it is a sumed the mass per unit area i I. Fir t one calculates the polar moment of inertia I p. The m,ass dm of the dark circular ring in the diagram is dm = 2ne de. so that I,. =

= 2'f f (P d(! = n r /2 = nd / 32. For reasons of symmetry. I" = I, o = 21.. , I" = 1,./2, so that the axial moment are Ix = I, = 1f-d 4 /64. 4


and therefore I p


j r/ dm I"

+ I,

20.4. Vector analysis


Steiner's theorem. Let Ie be the moment of inertia of a body about an axis C passing through the centre of mass. The moment of inertia I,f. about an axis A parallel to C and at a distance a from C is evidently I,f. = f (x + a)2 dm = f (x 2 + 2xa + a2) dm = Ie + 2a f x dm + a2m. m



But x denotes distance from the axis C, so that the integral

f x dm is the static moment about this


axis and is zero because the axis C passes through the centre of mass. Steiner's theorem. 1be molDent of inertia I A of a body about an arbitrary axis is equal to its moment 01 Inertia about the axis C through the centre of mass and pualJel to A pJus the product 01 its mass and the 8q1IIU"e of the distance from A to C. Steiner's theorem

20.4. Vector analysis In vector analysis one considers vector-valued functions of one or several variables and applies the concepts and methods of the differential and integral calculus. Its applications lie mainly in the fields of mathematical physics and of differential geometry. Fields Scalar fields. A scalar function rp in space is called a scalar field if, in a given region, a scalar rp(x, y, z) = rp(r) is assigned to each point P(x, y, z) with position vector r ; for instance, temperature or density in a body are scalar fields. They can be visualized through the level surfaces rp(x, y, z) = const in space or the level curves rp(x, y) = const in the plane; for example, there are maps giving lines of constant height above sea level (contours) and lines of constant temperature (isotherms). The function rp changes the more rapidly the closer the level surfaces or level curves are to each other. Example: The level surfaces of tbe scalar field rp = x2 + y2 + z2 = r2 are spberes with centre at the origin. Vector fields. Ifa function 11= a b II(x, y , z) = II(r) assigns a vector II to 'P° c • 2h each point in a region, then II = 11(,) "'P- c -h describes a vector field ; for instance, fields of force F(r) or electric fields E(r) "- \. are vector fields. These can be visualized by attaching arrows to different points ....... r whose lengths and directions represent the corresponding vector 11(,) (Fig.).



'20.4-1 a) Scalar field b) vector field

" \



\ I I I I \

One usually writes the vector functions of a vector field in the form 11= II(X, y, z, t) = u(x, y, z, t); + v(x, y, z, t)j + w(x, y, z, t) k, that is, the field vector II depends on the time t and on the space coordinates x, y, z, which may themselves be functions of I . Differentiation of this vector function is reduced to differentiation of the scalar functions u, v, w by the definition dll = du; + dv j + dw k, ou ou ou ou . . where du = dx + dy + dz + at dt With analogous expressIOns for dv and dw. An




equivalent definition is the following: ~ = lim II(X + Llx, y, z, t) - II(X, y, z, t) ox ..:1 ...... 0 Llx and analogously


= ~ ; + ~ . +~ k ox


ox J



oy oy oy oy' oz oz OZ J OZ ' That is to say, differentkltion is carried out componentwise according to the usual rules. For example, let III and III be vector functions and rp a scalar function; then the following rules for differentiating


20. Integral calculus

products hold:

~: + ~~

I) :x (lPa) = IP a ax ( a ( 3) ax 2)

411 •

az) =

411 X 412

a ay (411


411 •

a and similarly for


aaz aa1 . . ax + ax' 412 and Similarly

= 411 X

aa2 + -axaa1 x -ax-


Tz (411

X 412),


X 412),



aaz (lPa), a

:t (lPa);

a a ay (411 ' 412), Tz (411 ' 412), at (411 ' 412);

. '1 ar Iy ..lor an d simi

a at (411 X 412) '

In what follows, it will be assumed, to simplify matters, that all functions and the partial derivatives that occur are continuous, so that the order of partial differentiations may be interchanged; for



instance, ax ay = ayax (see Chapter 19.). The most important special cases of vector functions are: I. The field vector a does not depend explicitly on the time variable t and so has the form a(x, y, z) = u(x, y, z) i + v(x, y, z) j + w(x, y, z) k ; the field is then said to be time-independent. 2. The field depends on a scalar parameter t, not necessarily identical with the time variable; thus u;= x(t), v = y(l), W = z(t), or 41=

r(t) = x(t) i + y(t)j + z(t) k . = ,(t), as t varies, describes a path in space; if now t is the time variable and

The position vector r

,(t) the position of a particle, then the derivative


= :; i + :; j + :~


= xi + Yj + ik

represents the velocity of the particle and dd 2: its acceleration. The vector dd, is tangent to the t t d, path ,(1) (Fig.). If ,(1) has constant length, 1,1 = const, then also ,2 = const and hence"-d d, d,. t + dt . , = 0, so that' and dt are perpendicular. 20.4-2 Tangent vector

,-grad 'P

o¢:.._ _ _-'r'---_ _-.:{ 20.4-3 Level surfaces and gradient Gradient and potential Gradient. Consider the scalar function IP = IP(') = lP(x, y, z). If r is increased by

d, = dx i + dy j + dz k, then the change dIP in IP is given by dIP = ~~ dx + ~~ dy + ~~ dz . This expression can be regarded as the scalar product of the vector d" describing the small movement in space, and a vector grad IP


~ i + ~; j + ~~

k , the g,adient of IP. The change in the direc-

tion d, is then dIP = grad IP ' d,. If, in particular, one chooses d, so that it lies in a level surface IP = const, then IP does not change, so that dIP = 0 ; hence the gradient grad IP is perpendicular to the level surfacelP = const (Fig.). Now suppose that dr is at an angle D to grad IP ; then dIP = gradlP . dr = Igrad IPI . Id,1 cos D, and hence the function increases most rapidly when D = 0, that is, in the direction of the gradient. If one also sets Idrl = dS' 1 a a a then ~ = Igrad IPI cos D, which means: Gradient grad rp = i+ j + k




of rp in any direction is equal


a? a;- -t

the projection of the gradient onto this direction.

Potential. The operation of forming the gradient led from a scalar field IP = !p{r) to a vector field grad IP. In general, one cannot go in the opposite direction, that is, not every vector field is the gradient of a scalar field; vector fields a =

~~ i + ~~ j + ~~

conservative, and IP is called the potential of a (see Chapter 37.).

k which are gradients are called


20.4. Vector analysis p


= J(u dx + v dy + w dz), and let the path of integration Po r = r(/) = X(/) i + Y(/)j + z(r) k, so that the line integral can

Consider the line integral fa dr Po

given in the parametric form written as an ordinary integral

J(a ~: )dl J I

477 be






~~ + v(r) ~~ + w(/) ~~] d/.


In general, this integral depends not only on the end-points Po and P of the path of integration, but also on the whole path. However, if the vector field a = u(x, y, z) i + v(x, y , z) j + w(x, y, z) k and the path of integration r = x(r); + y(r)j + Z(/) k are defined in a simply-connected region G and have continuous derivatives there, then a = grad rp is a necessary and sufficient condition for the line integral to be independent of the path. Then the following theorem holds: The lioe integral of a consel'l'8tive vector does not depend on tbe path of integration, and Is equal to tbe potential cUtrerence between the Inltial and final points of tbe path. Conversely, if tbe line integral f a dr depends only on tbe end-points of the path, tben a is tbe gracUent of a potential fII. An equivalent statement is: In a simply-connected region G the line integral if> a dr vanishes for every closed path in G if and only if a = grad fII for some scalar function fII. A necessary and sufficient condition for a = grad

( - y dx + x dy) round the unit circle with centre at the origin. - The x dy) unit circle has the parametric representation x = cos I, Y = sin I . Hence if>(-y dx

+ 21r + cos I(COS/)] dr = f [sin 2 1 + COS2 /] dr = :In. The integral does not vanish, that is, the vector field a = - xi + y/is not conservative; this also follows from aau = - I, h Y whereas ~ = + I. =

1[- sin 21r

I(- Sin I)

Divergence and theorem of Gauss Divergence. The divergence is a scalar field that can be derived from a vector field Ii = u(x, y, z); + v(x, y, z)j + w(x, y, z) k. As an aid to interpretation one thinks of the field a as the velocilY field of a fluid flow of density e = e(x, y, z). One assumes that the flow of the fluid is sleady, that is, a and e do not depend explicitly on time. The x-component u; represents the distance per unit time in the x-direction covered by a fluid particle. If one considers a small cuboid (Fig.), with edges parallel to the axes, then the volume of fluid entering across the surface dAl = dy dz perpendicular to the x-axis in unit time is dAlu = u dy dz, and therefore the mass entering per unit time is eu dy dz. The mass leaving the y volume element dA2 = dy dz is

+ dx,y, z) u(x + dx, dy, dz) dy dz = [eu + a~eu) the difference a~~) dx dy dz gives the rate at whi~h


dX] dy dz; mass is lost

from the volume element across the faces dAl and dA z . The total loss per unit time is obtained by adding contributions from the other two pairs of opposite faces and is equal to

[a~~) + a~~) + a~:)]



dx dy dz.

The loss of mass per unit of lime and volume, the divergence of the vector field, is therefore given · a a f ter putttng . a = eQ ,- dIv. a = _au + _au + _aw b y d IV u = eu, v = ev, w = ew. ax ay az





20.4-4 Interpretation of diver-


If there is no loss or gain of mass, then diva = 0 and the field is called source-free. Points where diva> 0 are called sources (more mass flows out than in); points with diva 0 are called sinks (more flows in than out).

10, the curves can hardly be separated from each other when Ixl 0.6 but for x = 0.2, for instance, they differ quite appreciably (Fig.). That is to say: the index N beyond which the size of the remainder R,,(x) lies below a given positive number e depends, in general, on the choice of x within the interval I . These indices N(e, x), in the present example, increase indefinitely as x ...... 0 for any given e > O. If, however, one can find an N that does not depend on x, one speaks of uniform convergence of the function series F(x). This is always the case if the set of numbers N(e, x) is bounded above whenever e is fixed and x varies over 1. It will be shown in what follows that in this case F(x) is continuous, and also that the series F(x) can be integrated' term-by-term' and can be differentiated in this way if the functions f.(x) are all differentiable over I and the series resulting from term-by-term differentat ion is uniformly convergent. On the other hand, the series 1: x 2 (1 - x 2 )" considered earlier is convergent, but not uniformly convergent, in the interval - I ,,;:;;; x ,,;:;;; 1. x 2 (1


x 2 ), • • • (I - x 2 )


" '\





- 0.6





21.1-1 Approximal ions Fn(x ) to F(x)

A fUDction series F(x)

= E x'(1




- x')" for n




0, 1, 5. 10, 20,100


= E f,,(x) is called uniformly convergent in an interval I

e > 0, tbere exists an N IR,,(x)1 =

o 00

.-0 = N(e),

1/.+1 (x) + /.+2 (x) + ···1

if, for each given

depending only on e but not on x, such that < e for every x in I provided that " ;;. N .

The concept of uniform convergence was introduced by Karl WEJERSlRASS (1815-1897) and others; it can be extended, as can the following criterion, to the complex case. The Weierstrass majorant criterion (' M-test'), If each function/.(x) in tbe series F(x) = is bounded, with If,,(x)1

~ M"

for all x in I , aacl if tbe series



1: /.(x)


1: M" converges, tben tbe series


21.1. Series of functions F(x)





E In (x) is uniformJy convergent in the interval I . The series E Mn is then called a con-

11 · 0



vergent majorant for E In(x). n-O

Example J: The series E sin (nx)/n l is uniformly convergent for all x, because I in (lIx)/nl l~ I/n l ~


for aI/ x and the majorant series E l/n l converges . ex)

0 one can always find an NCe) such that for n > N(e) IR.(xo)1 = Ix'b+! + x'b+ 1 + ···1= x'b+I/(1 - xo) e. But if Xo increases towards + 1, then IR .. I increases indefinitely for any fixed n N(e), tbat is, lim x~ + J /(1 - xo) = 00. This shows that the


r; the interval from -r to +r is the range or interval of convergence (Fig.). One may put r = 00 for an always convergent series and r = 0 for a never convergent series. Theorem of Abel. For every power series that is neither always convergent nor never convergent, there exists an r 0 ucb that the series converges tor Ixl < r and diverges tor

COnyergMf dtyergenf

di yergenf



21.2-1 Convergence interval of a power series

Ixl> ',

The Cauchy-Hadamard formula. A formula for finding the radius of convergence was stated in 1821 by CAUCHY (1789-1857) but attracted no attention. It was only rediscovered 70 years

later by


(1865-1963). One considers the upper limit Il = lim "Vla,,1 of the sequence



1011, Vl a21, Vl a31, "', Vla"I,


that is, the number Il with the property that for every E 0 infinitely many terms of the sequence are greater than Il - E, but only finitely many are greater than Il + E. If Il is finite and positive, 0 < Il < +00, then I/Il is also finite and positive, and one can find Xl and e such that IXll < e < 1/1l, so that lIe> Il· This means that Vla,,1 < lIe or VlaAI < IXll/e < I for all Nl . The power series therefore converges absolutely at Xl' On the other


hand, if IXll > 1/Il, then "Vla,,1 > I/Ix21 or laAI > I for Radius of infinitely many n, so that the series diverges at Xl' Thus, convergence the number r = I/Il is the radius of convergence. The following result, stated without proof, sometimes " makes it possible to use the sequenec also converges and to the same limit . Example: The power series





Ex", E





x"/nl • ....




x"/n P (p ;;;' 0 fixed)

all have the same radius of convergence r = J. It suffices to calculate. for each p = O. 1.2, ...• the following limit: lim 10.+1 /0,,1= lim InP/(n + WI = lim [I - I /(n + IW = I . II~ OO

" ... 00


It is not possible to make any generally valid assertions about the behaviour of a power series at the ends X = +r and x = - r of the interval of convergence; a separate investigation must be

made in each particular case. For instance. in the first three series of the preceding example one finds that: 00

I. the series I x· diverges for x = - I and x = 11·1

2. the series

+I ;

:£ x"ln converges for x = - I and diverges for x = + I;

,,·1 00 3. the series I x·ln 2 converges for x ,,·1

= - I and x =

+ 1.


21. Series of fUDctions 00

H a power lel'ies E a,.r' bas the radJus of convergeace " then It converges absolutely for any x wltb Ixl ,,·0

< ,.

Uniform CODVergence of power series. A theorem due to ABEL states: A power leries converges uniformly In every closed Interval dult lies entirely inside the interval of coDvergence.

According to this theorem all results on series of functions obtained on the assumption of uniform convergence are valid for power series. Hence in every closed interval inside the interval of convergence f(x)




a"x" is a continuous function whose integral can be obtained by term-by-term

integration. Its derivative can be obtained by term-by-term differentiation. as will be shown later. Power series with a complex variable. For power series with complex cOefficients and a complex variable. the interval of convergence is replaced by a circular disc. whose radius is again called the radius of convergence (see Chapter 23.). Important properties of power series Identity theorem for power series. The power series f(x) its convergence interval Ixl

= E

< r. and in particular at x =

a,.r' is a continuous function inside 00 ".0 o. If the power series g(x) = 1: b"x" is


defined in the same interval. and hence continuous there, and if there is a sequence Xt with infinitely many non-zero-terms and x = 0 as an accumulation point. then it follows fromf(xt) = g(Xt) for all Xt and from limf(xt) ao, lim g(x,,) bo , that ao bo . Since x" =F 0, one can now consider


k .. co


k .. oo


two new functions fl(X,,) = (f(x,,) - ao)/x" = al a2X" a3xl gl(X,,) = (g(x,,) - bo)/x" = bl b2x" b3xl for which againfl(x,,) = gl(X,,), so that one obtains al = b l letting k -+ 00 . The procedure can be repeated to obtain a2 = b 2 , and by induction it follows that a" = b" for all n, so that the two power series are identical.

+ +

If the power series

of points

£ a,.r' and f


x" wltb Xl =F 0 and x"

+ +

+ ... , + ... .

b,.r' converge for Ixl

11.0 -+ 0,

< , aocllf their sums coincide on a sequence

then the series are Identkal, dull Is, a"

The identity theorem also holds for power series of the form

E a,,(x ".0

= b" for all II.

xo)". If a function f(x)

can be represented, in a neighbourhood of xo, by such a power series, then this representation is unique: if two methods of calculation lead to two power series representing a given function, then the coefficients of corresponding powers must be equal. The method of equating coefficients, which was derived in Chapter S. is therefore applicable to power series.




Example: For arbitrary real numbers a and b, (I x)" ( I X)b = (I x)"H. In the domain of convergence Ix l 1 each factor can be represented by the binomial series

'iti a, one needs to do no more than to write

Va = V[b" . (aW)) = b V[I + (a - b")/b"]; for instance, V33 = 2 V(I + 1/32). If this method is inappropriate, it may be possible to find a rational number whose nth power is near to a. A classical example is V2. Here V2>'iti 1.4 = 7/5, so that one can write n


V2 = Similarly 3 V92 =






VC:~ ~~/ )= ~


4.5 =


V( ~~)



V( 1+ :9)'


, so that one can write


9 3 . 2 3 . 92 ) 9 3 23 . 93 = 2'

V( +

736 ) 9 3 729 = 2'


7) 729 .

These examples indicate how the binomial series may be used to calculate roots very accurately. The series can also be used for any fractional exponent, and is particularly useful when 4-, 5- or even 7-place tables do not give enough accuracy. Approximations. In rough calculations it has proved advantageous to use the first few terms of a power series expansion. There are familiar applications of this in science and technology. To neglect the square and higher powers of small quantities is quite common, for example in thermodynamics, when the cubic coefficient of expansion is set to be three times the linear coefficient. Frequently sin x for small angles x is replaced by x. The table that follows gives a survey of the more frequently used formulae and their range of validity. The cited values of Ixl should not be exceeded if the error in using the approximation is not to exceed 0.001 or 0.01. It will be seen that for practical purposes it is frequently unnecessary to use tables of function values, since it is then often sufficient to keep the error below 0.1 % or 1 %. In that case it is, for example, permissible to replace arcsin x


21. Series of functions

by x in the range up to 10° ; this leads to a remarkable saving of effort. It is also possible to dispense with searching for tables of rare special functions if suitable approximate formulae can be derived from expansions in series; approximate calculation of integrals can also often be made from suitable approximation formulae quickly and with good accuracy. Ex amples:

V(2;~i )~

1:, 258.3 = 4


(I+ ~';56 )

4 4 = 4.009. The exact value to five places is 4.00895 (0.1)2 - 2 - = 0.905 (exact value 0.90484 ... ).

2. e- O• l I - 0.1 3. e- o.02J ~ I - 0.023 = 0.977 (exact value 0.977262 .. .).

Frequently used approximations. The figure at the bottom gives the relation between angles x in degrees and x in rad.

!~~roxima-I f~~~r ~ 10I


I /(1 + I /(1 + I /(1 + V(I +


x) X)l X)3 x)

V(J + x) V(J + x) /V(J + x) 3 N(J + x)


1 + x)/(I - x) (J + x)/(1 - xW V[(1 + x)/(1 - x)j sinx sin 2 x cos x cOS2 x t an x a resin x a rccos x a rctan x arccot x eJC In (1 + x) Ig (I + x) sinhx cosh x tanh x sinh- 1 x tanh- 1 x



for Ixl


I-x 1 - 2x 1 - 3x 1 + x /2

0.031 0.018 0.012 0.087

1 + x/3 1 + x/4 1 - x/2 1 - x/3 1 + 2x 1 +4x l+x x 0 1 1 x x n /2 - x x n/2 - x I+x x 0.4343x x I x x x

0.095 0.10 0.050 0.065 0.022 0.011 0.043 0.18 0.031 0.044 0.031 0.14 0.18 0.18 0.14 0.14 0.044 0.044 0.069 0.18 0.044 0.14 0.18 0.14

0.099 0.055 0.039 0.25 0.27 0.29 0.15 0.19 0.068 0.034 0.13 0.39 0.10 0.14 0.10 0.30 0.38 0.38 0.31 0.31 0.13 0.14 0.23 0.39 0.14 0.31 0.40 0.30

2nd I approximation 1 - x + Xl 1 - 2x + 3x 2 1 - 3x + 6x 2 1 + x /2 - x 2/8 I + x/3 - x 2/9 1 + x/4 - 3x 2/32 1 - x/2 + 3x 2/8 1 - x /3 + 2x 2/9 1 + 2x + 2X2 1 + 4x + 8x 2 1 + x + x 2 /2 x - x 3/6 x2 1 - x 2/2 1 - Xl x + x 3/3 x + x 3/6 n/2 - x - x 3/6 x - x 3/3 n/2 - x + x 3/3 I + x + x 2/2 x - x 2/2 0.4343x + 0.2171x2 x + x 3 /6 1+ x 2 /2 x - x 3 /3 x - x 3 /6 x + x 3/3

7. ., r .. 1· · ··?··· t.. I . f ... 1. . I 'I!"

o aDt . a02' ao] a'o/ aos' aOt;' a07 a08' a'og 0




0.'11 at2


Error ~ 3 10- 2 I 10for Ixl ~ 0.096 0.063 0.046 0.25 0.25 0.24 0.14 0.17 0.077 0.043 0.12 0.63 0.23 0.39 0.23 0.38 0.42 0.42 0.35 0.35 0.17 0.14 0.20 0.65 0.39 0.38 0.43 0.37

.l . . ~ . . I . r . . . .tp . l . 1J 0 a'14 0 a1S 0 a'16 0 a~7 ' a 18

0.20 0.12 0.095 0.48 0.47 0.49 0.28 0.34 0.16 0.090 0.25 1.04 0.41 0.70 0.42 0.58 0.63 0.63 0.57 0.57 0.38 0.33 0.45 1.03 0.70 0.61 0.70 0.52



Geometrical applications of Taylor's theorem Osculating parabola. Suppose that a curve has the equation y = 00 + 0lX + a2x2 + ." in a given coordinate system. If one introduces a new coordinate system in which the x-axis is tangent to the curve at a point P and the y-axis is along the normal at the same point, then for the curve referred to the new coordinate system the following values holds at the point P: x = 0, /(x) = 0, /,(x) = O. The curvature" is given, in general, by x = /"/(1 /,2)3/2, and therefore /"(0) = x. It follows that/(O) = 00 = 0,/'(0) = a1 = 0,/"(0) = 202 = x, and that the equation of the curve in the new coordinates is /(x) = (1/2) xx 2 + .... The curve near P is therefore approximated well


21.2. Power series by the parabola g(x) = (l/2) "xl, the osculating parabola. This is a second order approximation (Fig.). 21.2-4 Osculating parabola for the function y = I - cos x


", ",




", ",