Chapter Ⅵ Integral calculus in one variable 4
1 Jump continuous functions 4
Staircase and jump continuous functions 4
A characterization of jump continuous functions 6
The Banach space of jump continuous functions 7
2 Continuous extensions 10
The extension of uniformly continuous functions 10
Bounded linear operators 12
The continuous extension of bounded linear operators 15
3 The Cauchy-Riemann Integral 17
The integral of staircase functions 17
The integral of jump continuous functions 19
Riemann sums 20
4 Properties of integrals 25
Integration of sequences of functions 25
The oriented integral 26
Positivity and monotony of integrals 27
Componentwise integration 30
The first fundamental theorem of calculus 30
The indefinite integral 32
The mean value theorem for integrals 33
5 The technique of integration 38
Variable substitution 38
Integration by parts 40
The integrals of rational functions 43
6 Sums and integrals 50
The Bernoulli numbers 50
Recursion formulas 52
The Bernoulli polynomials 53
The Euler-Maclaurin sum formula 54
Power sums 56
Asymptotic equivalence 57
The Riemann ζ function 59
The trapezoid rule 64
7 Fourier series 67
The L2 scalar product 67
Approximating in the quadratic mean 69
Orthonormal systems 71
Integrating periodic functions 72
Fourier coefficients 73
Classical Fourier series 74
Bessel's inequality 77
Complete orthonormal systems 79
Piecewise continuously differentiable functions 82
Uniform convergence 83
8 Improper integrals 90
Admissible functions 90
Improper integrals 90
The integral comparison test for series 93
Absolutely convergent integrals 94
The majorant criterion 95
9 The gamma function 98
Euler's integral representation 98
The gamma function on C＼（-N） 99
Gauss's representation formula 100
The reflection formula 104
The logarithmic convexity of the gamma function 105
Stirling's formula 108
The Euler beta integral 110
Chapter Ⅶ Multivariable differential calculus 118
1 Continuous linear maps 118
The completeness of L（E,F） 118
Finite-dimensional Banach spaces 119
Matrix representations 122
The exponential map 125
Linear difierential equations 128
Gronwall's lemma 129
The variation of constants formula 131
Determinants and eigenvalues 133
Fundamental matrices 136
Second order linear differential equations 140
2 Differentiability 149
The definition 149
The derivative 150
Directional derivatives 152
Partial derivatives 153
The Jacobi matrix 155
A differentiability criterion 156
The Riesz representation theorem 158
The gradient 159
Complex differentiability 162
3 Multivariable differentiation rules 166
Linearity 166
The chain rule 166
The product rule 169
The mean value theorem 169
The differentiability of limits of sequences of functions 171
Necessary condition for local extrema 171
4 Multilinear maps 173
Continuous multilinear maps 173
The canonical isomorphism 175
Symmetric multilinear maps 176
The derivative of multilinear maps 177
5 Higher derivatives 180
Definitions 180
Higher order partial derivatives 183
The chain rule 185
Taylor's formula 185
Functions of m variables 186
Sufficient criterion for local extrema 188
6 Nemytskii operators and the calculus of variations 195
Nemytskii operators 195
The continuity of Nemytskii operators 195
The differentiability of Nemytskii operators 197
The differentiability of parameter-dependent integrals 200
Variational problems 202
The Euler-Lagrange equation 204
Classical mechanics 207
7 Inverse maps 212
The derivative of the inverse of linear maps 212
The inverse function theorem 214
Diffeomorphisms 217
The solvability of nonlinear systems of equations 218
8 Implicit functions 221
Differentiable maps on product spaces 221
The implicit function theorem 223
Regular values 226
Ordinary differential equations 226
Separation of variables 229
Lipschitz continuity and uniqueness 233
The Picard-Lindel？f theorem 235
9 Manifolds 242
Submanifolds of Rn 242
Graphs 243
The regular value theorem 243
The immersion theorem 244
Embeddings 247
Local charts and parametrizations 252
Change of charts 255
10 Tangents and normals 260
The tangential in Rn 260
The tangential space 261
Characterization of the tangential space 265
Differentiable maps 266
The differential and the gradient 269
Normals 271
Constrained extrema 272
Applications of Lagrange multipliers 273
Chapter Ⅷ Line integrals 281
1 Curves and their lengths 281
The total variation 281
Rectifiable paths 282
Differentiable curves 284
Rectifiable curves 286
2 Curves in Rn 292
Unit tangent vectors 292
Parametrization by arc length 293
Oriented bases 294
The Frenet n-frame 295
Curvature of plane curves 298
Identifying lines and circles 300
Instantaneous circles along curves 300
The vector product 302
The curvature and torsion of space curves 303
3 Pfaff forms 308
Vector fields and Pfaff forms 308
The canonical basis 310
Exact forms and gradient fields 312
The Poincaré lemma 314
Dual operators 316
Transformation rules 317
Modules 321
4 Line integrals 326
The definition 326
Elementary properties 328
The fundamental theorem of line integrals 330
Simply connected sets 332
The homotopy invariance of line integrals 333
5 Holomorphic functions 339
Complex line integrals 339
Holomorphism 342
The Cauchy integral theorem 343
The orientation of circles 344
The Cauchy integral formula 345
Analytic functions 346
Liouville's theorem 348
The Fresnel integral 349
The maximum principle 350
Harmonic functions 351
Goursat's theorem 353
The Weierstrass convergence theorem 356
6 Meromorphic functions 360
The Laurent expansion 360
Removable singularities 364
Isolated singularities 365
Simple poles 368
The winding number 370
The continuity of the winding number 374
The generalized Cauchy integral theorem 376
The residue theorem 378
Fourier integrals 379
References 387
Index 389