Chapter 1 Euclidean spaces 1
1.1 The real number system 2
1.2 Euclidean En 5
1.3 Elementary geometry of En 10
1.4 Basic topological notions in En 14
1.5 Convex sets 19
Chapter 2 Elementary topology of En 28
2.1 Functions 28
2.2 Limits and continuity of transformations 31
2.3 Sequences in En 37
2.4 Bolzano-Weierstrass theorem 43
2.5 Relative neighborhoods,continuous transformations 47
2.6 Topological spaces 50
2.7 Connectedness 56
2.8 Compactness 60
2.9 Metric spaces 62
2.10 Spaces of continuous functions 67
2.11 Noneuclidean norms on En 70
Chapter 3 Differentiation of real-valued functions 76
3.1 Directional and partial derivatives 76
3.2 Linear functions 79
3.3 Differentiable functions 82
3.4 Functions of class C(q) 89
3.5 Relative extrema 99
3.6 Convex and concave functions 107
Chapter 4 Vector-valued functions of several variables 119
4.1 Linear transformations 119
4.2 Affine transformations 125
4.3 Differentiable transformations 128
4.4 Composition 134
4.5 The inverse function theorem 140
4.6 The implicit function theorem 147
4.7 Manifolds 153
4.8 The multiplier rule 161
Chapter 5 Integration 167
5.1 Intervals 168
5.2 Measure 170
5.3 Integrals over En 181
5.4 Integrals over bounded sets 186
5.5 Iterated integrals 190
5.6 Integrals of continuous functions 200
5.7 Change of measure under affine transformations 206
5.8 Transformation of integrals 209
5.9 Coordinate systems in En 216
5.10 Measurable sets and functions;further properties 222
5.11 Integrals：general definition,convergence theorems 227
5.12 Differentiation under the integral sign 237
5.13 Lp-spaces 240
Chapter 6 Curves and line integrals 245
6.1 Derivatives 245
6.2 Curves in En 247
6.3 Differential 1-forms 253
6.4 Line integrals 258
6.5 Gradient method 265
6.6 Integrating factors;thermal systems 268
Chapter 7 Exterior algebra and differential calculus 275
7.1 Covectors and differential forms of degree 2 276
7.2 Alternating multilinear functions 283
7.3 Multicovectors 287
7.4 Differential forms 291
7.5 Multivectors 295
7.6 Induced linear transformations 306
7.7 Transformation law for differential forms 309
7.8 The adjoint and codifferential 311
7.9 Special results for n=3 316
7.10 Integrating factors(continued) 318
Chapter 8 Integration on manifolds 321
8.1 Regular transformations 322
8.2 Coordinate systems on manifolds 329
8.3 Measure and integration on manifolds 334
8.4 The divergence theorem 340
8.5 Fluid flow 350
8.6 Orientations 353
8.7 Integrals of r-forms 356
8.8 Stokess formula 362
8.9 Regular transformations on submanifolds 367
8.10 Closed and exact differential forms 369
8.11 Motion ofa particle 375
8.12 Motion of several particles 380
Appendix 1 Axioms for a vector space 383
Appendix 2 Mean value theorem;Taylors theorem 385
Appendix 3 Review of Riemann integration 386
Appendix 4 Monotone functions 388
References 389
Answers to problems 391
Index 405