Analysis II

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ISBN: 9783764371050 | 3764371056
Cover: Paperback
Copyright: 3/18/2006

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The second volume of this introduction into analysis deals with the integration theory of functions of one variable, the multidimensional differential calculus and the theory of curves and line integrals. The modern and clear development that started in Volume I (3-7643-7153-6) continues. In this way a sustainable basis will be created which allows to deal with interesting applications that sometimes go considerably beyond the material that is represented in traditional textbooks. This applies, for instance, to the exploration of Nemytskii operators which enable a transparent introduction into the calculus of variations and the derivation of the Euler-Lagrange equations. Another example is the presentation of the local theory of submanifolds of Rn.

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

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Table of Contents

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