Functions of Several Variables

The purpose of this book is to give a systematic development of differential and integral calculus for functions of several variables. The traditional topics from advanced calculus are included: maxima and minima, chain rule, implicit function theorem, multiple integrals, divergence and Stokes's theorems, and so on. However, the treatment differs in several important respects from the traditional one. Vector notation is used throughout, and the distinction is maintained between n-dimensional euclidean space En and its dual. The elements of the Lebesgue theory of integrals are given. In place of the traditional vector analysis in £3, we introduce exterior algebra and the calculus of exterior differential forms. The formulas of vector analysis then become special cases of formulas about differential forms and integrals over manifolds lying in P. The book is suitable for a one-year course at the advanced undergraduate level. By omitting certain chapters, a one semester course can be based on it. For instance, if the students already have a good knowledge of partial differentiation and the elementary topology of P, then substantial parts of Chapters 4, 5, 7, and 8 can be covered in a semester. Some knowledge of linear algebra is presumed. However, results from linear algebra are reviewed as needed (in some cases without proof). A number of changes have been made in the first edition. Many of these were suggested by classroom experience. A new Chapter 2 on elementary topology has been added.

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.