Free E-Book: Advance Calculus

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Textbook: Advanced Calculus ©1990, 592 pages
Lynn H. Loomis / Shlomo Sternberg // Harvard University // PDE

Table of Content:

CONTENTS
Chapter 0 Introduction
1 Logic: 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 COlnposition 14
10 Duality 15
11 The Boolean operations. 17
12 Partitions and equivalence relations 19

Chapter 1 Vector Spaces
1 Fundalnental notions 21
2 Vector spaces and geometry 36
3 Product spaces and HomeV, 1fT) 43
4 Affine subspaces and quotient spaces 52
5 Direct sums 56
6 Bilinearity 67

Chapter 2 Finite-Dill1ensional Vector Spaces
1 Bases 71
2 Din1ension 77
3 The dual space 81
4 l\'1atrices 88
5 Trace and determinant 99
6 l\'1atrix computations 102
7 The diagonalization of a quadratic forn1 111

Chapter 3 The Differential Calculus
1 Review in ~ 117
2 Norlns . 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 IR n
• 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
*2 Topology .
3 Sequential convergence .
4 Sequential compactness.
5 Compactness and uniformity
6 Equicontinuity
7 Completeness.
8 A first look at Banach algebras
9 The contraction mapping fixed-point theorem
10 The integral of a parametrized arc
11 The complex number system
*12 Weak methods

Chapter 5 Scalar Product Spaces
1 Scalar products
2 Orthogonal projection
3 Self-adjoint transformations
4 Orthogonal transformations
5 Compact transformations

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 an 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 383
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 [3 . 459
Chapter 12 Potential Theory in lEn
1 Solid angle 474
2 Green's formulas . 476
3 The maximum principle 477
4 Green's functions 479
5 The Poisson integral formula
6 Consequences of the Poisson integral formula
7 Harnack's theorem
8 Subharmonic functions .
9 Dirichlet's problem .
10 Behavior near the boundary
11 Dirichlet's principle .
12 Physical applications
13 Problem set: The calculus of residues .

Chapter 13 Classical Mechanics
1 The tangent and cotangent bundles
2 Equations of variation .
3 The fundamental linear differential form on T* (M)
4 The fundamental exterior two-form on T*(M)
5 Hamiltonian mechanics .
6 The central-force problem .
7 The two-body problem .
8 Lagrange's equations
9 Variational principles
10 Geodesic coordinates
11 Euler's equations
12 Rigid-body motion
13 Small oscillations
14 Small oscillations (continued) .
15 Canonical transformations .

Selected References .
Notation Index
Index .

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