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Advanced Calculus

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TOC

Chapt 0 Introduction
1. Logic : quantifiers
2. The logical connectives
3. Negations aof quantifiers
4. Sets
5. Restricted variables
6. Ordered pairs and relations
7. Functions and mappings
8. Product sets, index notation
9. Composition
10. Duality
11. The boolean operations
12. Partitions and equivalence relations

Chapt 1 Vector Spaces
1. Fundamental notions
2. Vector spaces and geometry
3. Produc spaces and Hom(V,W)
4. Affine subspaces and quotient spaces
5. Direct sums
6. Bilinearity

Chapt 2 Finite-Dimensional Vector Spaces
1. Bases
2. Dimensions
3. THe dual space
4. Matrices
5. Trace and determinant
6. Matrix computations
7. The diagonalization of a quadratic form

Chapt 3 The Differential Calculus
1. Review in R
2. Norms
3. Continuity
4. Eqivalent norms
5. Infinitesimals
6. The differential
7. Directional derivatives, the mean-value theorem
8. The differential and product spaces
9. The differential and R
10. Elementary applications
11. The implicit-function theorem
12. Submanifolds and Lagrange multipliers
13. Functional dependence
14. Uniform continuity and function-valued mappings
15. The calculus of variations
16. The second differential and the classification of cri
17. The Taylor formula

Chapt 4. Compactness and Completeness
1. Metric spaces; open and closed sets
2. Topology
3. Sequential convergence
4. Sequential compactness
5. Compactness aand uniformity
6. Equicontinuity
7. Completeness
8. A first look at Banach algebras
9. The contraction mapping fixed-point theorem
10. The integral of parametrized arc
11. The complex number system

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

Chapt 6 Differential Equations
1. The fundamental theorem
2. Differentiable dependence on parameters
3. The linear equation
4. The nth-order linear eqation
5. Solving the inhomogeneous equation
6. The boundary-value problem
7. Fourier series

Chapt 7 Multilinear Functionals
1. Bilinear functionals
2. Multilinear functionals
3. Permutations
4. The sign of a permutation
5. The subspace of alternating tensors
6. The determinant
7. The exterior algebra
8. Exterior powers of scalar product spaces
9. The star operator

Chapt 8 Integration
1. Introduction
2. Axioms
3. Rectangles and paved sets
4. The minimal theory
6. Contented sets
7. When is a set contented?
8. Behavior under linear distortions
9. Axioms for integration
10. Integration of contented functions
11. The change of variables formula
12. Successive integration
13. Absolutely integrable functions
14. Problem set : The Fourier transform

Chapt 9 Differentiable Manifolds
1. Atlases
2. Functions, convergence
3. Differentiable manifolds
4. The tangent space
5. Flows and vector fields
6. Lie derivatives
7. Linear differential forms
8. Computations with coordinates
9. Riemann metrics

Chapt 10 The integral Calculus on Manifolds
1. Compactness
2. Partitions of unity
3. Densities
4. Volume density of a Riemann metric
5. Pullback and Lie derivatives of densities
6. The divergence theorem
7. More complicated domains

Chapt 11 Exterior Calculus
1. Exterior differential forms
2. Oriented manifolds and the integration of exterior differential forms
3. The operator d
4. Stokes theorem
5. Some illustrations of stokes theorem
6. The Lie derivative of a differential form

Chapt 12 Potential Theory in E
1. Solid angle
2. Greens’s formulas
3. The maximum principle
4. Green’s functions
5. The poisson integral formula
6. Consequences of the Poisson integral formula
7. Harnack’s theorem
8. Subharmonic functions
9. Dirichlet’s principle
12. Physical appliacations
13. Problem set : The calulus of residues

Chapt 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
15. Canonical transformations

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