Vector and Geometric Calculus (Geometric Algebra & Calculus) 1480132454, 9781480132450

This textbook for the undergraduate vector calculus course presents a unified treatment of vector and geometric calculus

516 117 7MB

English Pages 211 [186] Year 2012

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Contents
Preface
I Preliminaries
1 Curve and Surface Representations
1.1 Curve Representations
1.2 Surface Representations
1.3 Polar, Cylindrical, Spherical Coordinates
2 Limits and Continuity
2.1 Open and Closed Sets
2.2 Limits
2.3 Continuity
II Derivatives
3 The Differential
3.1 The Partial Derivative
3.2 The Taylor Expansion
3.3 The Differential
3.4 The Chain Rule
3.5 The Directional Derivative
3.6 Inverse and Implicit Functions
4 Tangent Spaces
4.1 Manifolds
4.2 Tangent Spaces to Curves
4.3 Tangent Spaces to Surfaces
5.1 Fields
5.3 Scalar and Vector Fields
5.4 Curvilinear Coordinates
5.5 The Vector Derivative
6 Extrema
6.1 Extrema
6.2 Constrained Extrema
III Integrals
7 Integrals over Curves
7.1 The Scalar Integral
7.2 The Path Integ ral
7.3 The Line Integral
7.4 Conservative Vector Fields
8 Multiple Integrals
8.1 Multiple Integrals
8.2 Change of Variables
9 Integrals over Surfaces
9.1 The Surface Integral
9.2 The Flux Integral
IV The Fundamental Theorem of Calculus
10.1 The Fundamental Theorem of Calculus
10.2 The Divergence Theorem
10.3 The Curl Theorem
10.4 Analytic Functions
V Differential Geometry
11 Differential Geometry in R³
11.1 Curves
11.2 Surfaces
11.3 Curves in Surfaces
VI Appendices
A Review of Geometric Algebra
B Software
C Formulas
D Differential Forms
Index

Citation preview

V ector and G eom etric Calculus A la n M a c d o n a ld Luther College, Decorah, IA 52101 USA [email protected] faculty.luther.edu / "macdonal