Author(s): Alan Macdonald
Year: 2012
Language: English
Pages: 186
Tags: Математика;Линейная алгебра и аналитическая геометрия;
Contents......Page 3
Preface......Page 6
I Preliminaries......Page 11
1 Curve and Surface Representations......Page 12
1.1 Curve Representations......Page 14
1.2 Surface Representations......Page 16
1.3 Polar, Cylindrical, Spherical Coordinates......Page 20
2.1 Open and Closed Sets......Page 22
2.2 Limits......Page 24
2.3 Continuity......Page 27
II Derivatives......Page 30
3.1 The Partial Derivative......Page 31
3.2 The Taylor Expansion......Page 36
3.3 The Differential......Page 38
3.4 The Chain Rule......Page 42
3.5 The Directional Derivative......Page 47
3.6 Inverse and Implicit Functions......Page 49
4.1 Manifolds......Page 54
4.2 Tangent Spaces to Curves......Page 56
4.3 Tangent Spaces to Surfaces......Page 60
5.1 Fields......Page 64
5.2 The Gradient......Page 65
5.3 Scalar and Vector Fields......Page 72
5.4 Curvilinear Coordinates......Page 76
5.5 The Vector Derivative......Page 82
6.1 Extrema......Page 84
6.2 Constrained Extrema......Page 89
III Integrals......Page 92
7.1 The Scalar Integral......Page 93
7.2 The Path Integ ral......Page 97
7.3 The Line Integral......Page 101
7.4 Conservative Vector Fields......Page 105
8.1 Multiple Integrals......Page 114
8.2 Change of Variables......Page 120
9.1 The Surface Integral......Page 123
9.2 The Flux Integral......Page 125
IV The Fundamental Theorem of Calculus......Page 129
10.1 The Fundamental Theorem of Calculus......Page 130
10.2 The Divergence Theorem......Page 135
10.3 The Curl Theorem......Page 139
10.4 Analytic Functions......Page 144
V Differential Geometry......Page 146
11.1 Curves......Page 147
11.2 Surfaces......Page 152
11.3 Curves in Surfaces......Page 159
VI Appendices......Page 167
A Review of Geometric Algebra......Page 168
B Software......Page 172
C Formulas......Page 179
D Differential Forms......Page 181
Index......Page 183
