Advanced Calculus

Preface......Page p0003.djvu
Contents......Page p0007.djvu
1 Logic: quantifiers......Page p0013.djvu
2 The logical connectives......Page p0015.djvu
4 Sets......Page p0018.djvu
5 Restricted variables......Page p0020.djvu
6 Ordered pairs and relations......Page p0021.djvu
7 Functions and mappings......Page p0022.djvu
8 Product sets; index notation......Page p0024.djvu
9 Composition......Page p0026.djvu
10 Duality......Page p0027.djvu
11 The Boolean operations......Page p0029.djvu
12 Partitions and equivalence relations......Page p0031.djvu
1 Fundamental notions......Page p0033.djvu
2 Vector spaces and geometry......Page p0048.djvu
3 Product spaces and Hom(V, W)......Page p0055.djvu
4 Affine subspaces and quotient spaces......Page p0064.djvu
5 Direct sums......Page p0068.djvu
6 Bilinearity......Page p0079.djvu
1 Bases......Page p0083.djvu
2 Dimension......Page p0089.djvu
3 The dual space......Page p0093.djvu
4 Matrices......Page p0100.djvu
5 Trace and determinant......Page p0111.djvu
6 Matrix computations......Page p0114.djvu
7 The diagonalization of a quadratic form......Page p0123.djvu
1 Review in R......Page p0129.djvu
2 Norms......Page p0133.djvu
3 Continuity......Page p0138.djvu
4 Equivalent norms......Page p0144.djvu
5 Infinitesimals......Page p0148.djvu
6 The differential......Page p0152.djvu
7 Directional derivatives; the mean-value theorem......Page p0158.djvu
8 The differential and product spaces......Page p0164.djvu
9 The differential and R^n......Page p0168.djvu
10 Elementary applications......Page p0173.djvu
11 The implicit-function theorem......Page p0176.djvu
12 Submanifolds and Lagrange multipliers......Page p0184.djvu
*13 Functional dependence......Page p0187.djvu
*14 Uniform continuity and function-valued mappings......Page p0191.djvu
*15 The calculus of variations......Page p0194.djvu
*16 The second differential and the classification of critical points......Page p0198.djvu
*17 The Taylor formula......Page p0203.djvu
1 Metric spaces; open and closed sets......Page p0207.djvu
*2 Topology......Page p0213.djvu
3 Sequential convergence......Page p0214.djvu
4 Sequential compactness......Page p0217.djvu
5 Compactness and uniformity......Page p0222.djvu
6 Equicontinuity......Page p0227.djvu
7 Completeness......Page p0228.djvu
8 A first look at Banach algebras......Page p0235.djvu
9 The contraction mapping fixed-point theorem......Page p0240.djvu
10 The integral of a parametrized arc......Page p0248.djvu
11 The complex number system......Page p0252.djvu
*12 Weak methods......Page p0257.djvu
1 Scalar products......Page p0260.djvu
2 Orthogonal projection......Page p0264.djvu
3 Self-adjoint transformations......Page p0269.djvu
4 Orthogonal transformations......Page p0274.djvu
5 Compact transformations......Page p0276.djvu
1 The fundamental theorem......Page p0278.djvu
2 Differentiable dependence on parameters......Page p0286.djvu
3 The linear equation......Page p0288.djvu
4 The nth-order linear equation......Page p0293.djvu
5 Solving the inhomogeneous equation......Page p0300.djvu
6 The boundary-value problem......Page p0306.djvu
7 Fourier series......Page p0313.djvu
1 Bilinear functionals......Page p0317.djvu
2 Multilinear functionals......Page p0318.djvu
3 Permutations......Page p0320.djvu
4 The sign of a permutation......Page p0321.djvu
5 The subspace A^n of alternating tensors......Page p0322.djvu
6 The determinant......Page p0324.djvu
7 The exterior algebra......Page p0328.djvu
8 Exterior powers of scalar product spaces......Page p0331.djvu
9 The star operator......Page p0332.djvu
1 Introduction......Page p0333.djvu
2 Axioms......Page p0334.djvu
3 Rectangles and paved sets......Page p0336.djvu
4 The minimal theory......Page p0339.djvu
5 The minimal theory (continued)......Page p0340.djvu
6 Contented sets......Page p0343.djvu
7 When is a set contented?......Page p0345.djvu
8 Behavior under linear distortions......Page p0347.djvu
9 Axioms for integration......Page p0348.djvu
10 Integration of contented functions......Page p0350.djvu
11 The change of variables formula......Page p0354.djvu
12 Successive integration......Page p0358.djvu
13 Absolutely integrable functions......Page p0363.djvu
14 Problem set: The Fourier transform......Page p0367.djvu
1 Atlases......Page p0376.djvu
2 Functions, convergence......Page p0379.djvu
3 Differentiable manifolds......Page p0381.djvu
4 The tangent space......Page p0385.djvu
5 Flows and vector fields......Page p0388.djvu
6 Lie derivatives......Page p0395.djvu
7 Linear differential forms......Page p0402.djvu
8 Computations with coordinates......Page p0405.djvu
9 Riemann metrics......Page p0409.djvu
1 Compactness......Page p0415.djvu
2 Partitions of unity......Page p0417.djvu
3 Densities......Page p0420.djvu
4 Volume density of a Riemann metric......Page p0423.djvu
5 Pullback and Lie derivatives of densities......Page p0428.djvu
6 The divergence theorem......Page p0431.djvu
7 More complicated domains......Page p0436.djvu
1 Exterior differential forms......Page p0441.djvu
2 Oriented manifolds and the integration of exterior differential forms......Page p0445.djvu
3 The operator d......Page p0450.djvu
4 Stokes\' theorem......Page p0454.djvu
5 Some illustrations of Stokes\' theorem......Page p0461.djvu
6 The Lie derivative of a differential form......Page p0464.djvu
Appendix I \"Vector analysis\"......Page p0469.djvu
Appendix II Elementary differential geometry of surfaces in E^3......Page p0471.djvu
1 Solid angle......Page p0486.djvu
2 Green\'s formulas......Page p0488.djvu
3 The maximum principle......Page p0489.djvu
4 Green\'s functions......Page p0491.djvu
5 The Poisson integral formula......Page p0494.djvu
6 Consequences of the Poisson integral formula......Page p0497.djvu
7 Harnack\'s theorem......Page p0499.djvu
8 Subharmonic functions......Page p0501.djvu
9 Dirichlet\'s problem......Page p0503.djvu
10 Behavior near the boundary......Page p0507.djvu
11 Dirichlet\'s principle......Page p0511.djvu
12 Physical applications......Page p0513.djvu
13 Problem set: The calculus of residues......Page p0515.djvu
1 The tangent and cotangent bundles......Page p0523.djvu
2 Equations of variation......Page p0525.djvu
3 The fundamental linear differential form on T*(M)......Page p0527.djvu
4 The fundamental exterior two-form on T*(M)......Page p0529.djvu
5 Hamiltonian mechanics......Page p0532.djvu
6 The central-force problem......Page p0533.djvu
7 The two-body problem......Page p0540.djvu
8 Lagrange\'s equations......Page p0542.djvu
9 Variational principles......Page p0544.djvu
10 Geodesic coordinates......Page p0549.djvu
11 Euler\'s equations......Page p0553.djvu
12 Rigid-body motion......Page p0556.djvu
13 Small oscillations......Page p0563.djvu
14 Small oscillations (continued)......Page p0565.djvu
15 Canonical transformations......Page p0570.djvu
Selected References......Page p0581.djvu
Notation Index......Page p0584.djvu
Index......Page p0587.djvu