Advanced Calculus (Revised Edition) available in Hardcover, Paperback
Advanced Calculus (Revised Edition)
- ISBN-10:
- 9814583928
- ISBN-13:
- 9789814583923
- Pub. Date:
- 03/12/2014
- Publisher:
- World Scientific Publishing Company, Incorporated
- ISBN-10:
- 9814583928
- ISBN-13:
- 9789814583923
- Pub. Date:
- 03/12/2014
- Publisher:
- World Scientific Publishing Company, Incorporated
Advanced Calculus (Revised Edition)
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Product Details
ISBN-13: | 9789814583923 |
---|---|
Publisher: | World Scientific Publishing Company, Incorporated |
Publication date: | 03/12/2014 |
Edition description: | Revised |
Pages: | 596 |
Product dimensions: | 6.60(w) x 9.80(h) x 1.50(d) |
Table of Contents
Chapter 0 Introduction
1 Logie: quantifiers 1
2 The logical connectives 3
3 Negations of quantifiers 6
4 Sets 6
5 Restricted variables 8
6 Ordered pairs and relations 9
7 Functions and mappings 10
8 Product sets; index notation 12
9 Composition 14
10 Duality 15
11 The Boolean operations 17
12 Partitions and equivalence relations 10
Chapter 1 Vector Spaces
1 Fundamental notions 21
2 Vector spaces and geometry 36
3 Product spaces and Hom(V, W) 43
4 Affine subspaces and quotient spaces 52
5 Direct sums 56
6 Bilinearity 67
Chapter 2 Finite-Dimensional Vector Spaces
1 Base 71
2 Dimension 77
3 The dual space 81
4 Matrices 88
5 Trace and determinant 99
6 Matrix computations 102
*7 The diagonalization of a quadratic form 111
Chapter 3 The Differential Calculus
1 Review in R 117
2 Norms 121
3 Continuity 126
4 Equivalent norms 132
5 Infinitesimals 136
6 The differential 140
7 Directional derivatives; the mean-value theorem 146
8 The differential and product spaces 152
9 The differential and Rn 156
10 Elementary applications 161
11 The implicit-function theorem 164
12 Submanifolds and Lagrange multipliers 172
*13 Functional dependence 175
*14 Uniform continuity and function-valued mappings 179
*15 The calculus of variations 182
*16 The second differential and the classification of critical points 186
*17 The Taylor formula 191
Chapter 4 Compactness and Completeness
1 Metric spaces; open and closed sets 196
*2 Topology 201
3 Sequential convergence 202
4 Sequential compactness 205
5 Compactness and uniformity 210
6 Equicontinuity 215
7 Completeness 216
8 A first look at Banach algebras 223
9 The contraction mapping fixed-point theorem 228
10 The integral of a parametrizod arc 236
11 The complex number system 240
*12 Weak methods 245
Chapter 5 Scalar Product Spaces
1 Scalar products 248
2 Orthogonal projection 252
3 Self-adjoint transformations 257
4 Orthogonal transformations 262
5 Compact transformations 264
Chapter 6 Differential Equations
1 The fundamental theorem 266
2 Differentiable dependence on parameters 274
3 The linear equation 276
4 The nth-order linear equation 281
5 Solving the inhomogeneous equation 288
6 The boundary-value problem 294
7 Fourier series 301
Chapter 7 Multilinear Functionals
1 Bilinear functionals 305
2 Multilinear functionals 306
3 Permutations 308
4 The sign of a permutation 309
5 The subspace Rn of alternating tensors 310
6 The determinant 312
7 The exterior algebra 316
8 Exterior powers of scalar product spaces 319
9 The star operator 320
Chapter 8 Integration
1 Introduction 321
2 Axioms 322
3 Rectangles and paved sets 324
4 The minimal theory 327
5 The minimal theory (continued) 328
6 Contented sets 331
7 When is a set contented? 333
8 Behavior under linear distortions 335
9 Axioms for integration 336
10 Integration of contented functions 338
11 The change of variables formula 342
12 Successive integration 346
13 Absolutely integrable functions 351
14 Problem set: The Fourier transform 355
Chapter 9 Differentiable Manifolds
1 Atlases 364
2 Functions, convergence 367
3 Differentiable manifolds 369
4 The tangent space 373
5 Flows and vector fields 376
6 Lie derivatives 384
7 Linear differential forms 390
8 Computations with coordinates 393
9 Riemann metrics 397
Chapter 10 The Integral Calculus on Manifolds
1 Compactness 403
2 Partitions of unity 405
3 Densities 408
4 Volume density of a Riemann metric 411
5 Pullback and Lie derivatives of densities 416
6 The divergence theorem 419
7 More complicated domains 424
Chapter 11 Exterior Calculus
1 Exterior differential forms 429
2 Oriented manifolds and the integration of exterior differential forms 433
3 The operator d 438
4 Stokes' theorem 442
5 Some illustrations of Stokes' theorem 449
6 The Lie derivative of a differential form 452
Appendix I "Vector analysis" 457
Appendix II Elementary differential geometry of surfaces in E3 459
Chapter 12 Potential Theory in En
1 Solid angle 474
2 Green's formulas 476
3 The maximum principle 477
4 Green's functions 479
5 The Poisson integral formula 482
6 Consequences of the Poisson integral formula 485
7 Harnack's theorem 487
8 Subharmonic functions 489
9 Dirichlet's problem 491
10 Behavior near the boundary 495
11 Dirichlet's principle 499
12 Physical applications 500
13 Problem set: The calculus of residues 503
Chapter 13 Classical Mechanic
1 The tangent and cotangent bundles 511
2 Equations of variation 513
3 The fundamental linear differential form on T*(M) 516
4 The fundamental exterior two-form on T*(M) 517
5 Hamiltonian mechanics 520
6 The central-force problem 523
7 The two-body problem 528
8 Lagrange's equations 530
9 Variational principles 532
10 Geodesic coordinates 537
11 Euter's equations 541
12 Rigid-body motion 544
13 Small oscillations 551
14 Small oscillations (continued) 553
15 Canonical transformations 558
Selected Reference 569
Notation Index 572
Index 575