Functions of Several Variables

1 Euclidean spaces. - 1. 1 The real number system. - 1. 2 Euclidean En. - 1. 3 Elementary geometry of En. - 1. 4 Basic topological notions in En. - *1. 5 Convex sets. - 2 Elementary topology of En. - 2. 1 Functions. - 2. 2 Limits and continuity of transformations. - 2. 3 Sequences in En. - 2. 4 Bolzano-Weierstrass theorem. - 2. 5 Relative neighborhoods continuous transformations. - 2. 6 Topological spaces. - 2. 7 Connectedness. - 2. 8 Compactness. - 2. 9 Metric spaces. - 2. 10 Spaces of continuous functions. - *2. 11 Noneuclidean norms on En. - 3 Differentiation of real-valued functions. - 3. 1 Directional and partial derivatives. - 3. 2 Linear functions. - **3. 3 Difierentiable functions. - 3. 4 Functions of class C(q). - 3. 5 Relative extrema. - *3. 6 Convex and concave functions. - 4 Vector-valued functions of several variables. - 4. 1 Linear transformations. - 4. 2 Affine transformations. - 4. 3 Differentiable transformations. - 4. 4 Composition. - 4. 5 The inverse function theorem. - 4. 6 The implicit function theorem. - 4. 7 Manifolds. - 4. 8 The multiplier rule. - 5 Integration. - 5. 1 Intervals. - 5. 2 Measure. - 5. 3 Integrals over En. - 5. 4 Integrals over bounded sets. - 5. 5 Iterated integrals. - 5. 6 Integrals of continuous functions. - 5. 7 Change of measure under affine transformations. - 5. 8 Transformation of integrals. - 5. 9 Coordinate systems in En. - 5. 10 Measurable sets and functions; further properties. - 5. 11 Integrals: general definition convergence theorems. - 5. 12 Differentiation under the integral sign. - 5. 13 Lp-spaces. - 6 Curves and line integrals. - 6. 1 Derivatives. - 6. 2 Curves in En. - 6. 3 Differential 1-forms. - 6. 4 Line integrals. - *6. 5 Gradient method. - *6. 6 Integrating factors; thermal systems. - 7 Exterior algebra and differential calculus. - 7. 1 Covectors and differential forms of degree 2. - 7. 2 Alternating multilinearfunctions. - 7. 3 Multicovectors. - 7. 4 Differential forms. - 7. 5 Multivectors. - 7. 6 Induced linear transformations. - 7. 7 Transformation law for differential forms. - 7. 8 The adjoint and codifferential. - *7. 9 Special results for n = 3. - *7. 10 Integrating factors (continued). - 8 Integration on manifolds. - 8. 1 Regular transformations. - 8. 2 Coordinate systems on manifolds. - 8. 3 Measure and integration on manifolds. - 8. 4 The divergence theorem. - *8. 5 Fluid flow. - 8. 6 Orientations. - 8. 7 Integrals of r-forms. - 8. 8 Stokes's formula. - 8. 9 Regular transformations on submanifolds. - 8. 10 Closed and exact differential forms. - 8. 11 Motion of a particle. - 8. 12 Motion of several particles. - Axioms for a vector space. - Mean value theorem; Taylor's theorem. - Review of Riemann integration. - Monotone functions. - References. - Answers to problems. Language: English