Functions of Several Variables

ISBN-13: 9780412233401 / Angielski / Miękka / 1981 / 138 str.

B. D. Graven; B. Craven

Opis

This book is aimed at mathematics students, typically in the second year of a university course. The first chapter, however, is suitable for first-year students. Differentiable functions are treated initially from the standpoint of approximating a curved surface locally by a fiat surface. This enables both geometric intuition, and some elementary matrix algebra, to be put to effective use. In Chapter 2, the required theorems - chain rule, inverse and implicit function theorems, etc- are stated, and proved (for n variables), concisely and rigorously. Chapter 3 deals with maxima and minima, including problems with equality and inequality constraints. The chapter includes criteria for discriminating between maxima, minima and saddlepoints for constrained problems; this material is relevant for applications, but most textbooks omit it. In Chapter 4, integration over areas, volumes, curves and surfaces is developed, and both the change-of-variable formula, and the Gauss-Green-Stokes set of theorems are obtained. The integrals are defined with approximative sums (ex- pressed concisely by using step-functions); this preserves some geometrical (and physical) concept of what is happening. Consequent on this, the main ideas of the 'differential form' approach are presented, in a simple form which avoids much of the usual length and complexity. Many examples and exercises are included.

Spis treści

1. Differentiable Functions.- 1.1 Introduction.- 1.2 Linear part of a function.- 1.3 Vector viewpoint.- 1.4 Directional derivative.- 1.5 Tangent plane to a surface.- 1.6 Vector functions.- 1.7 Functions of functions.- 2. Chain Rule and Inverse Function Theorem.- 2.1 Norms.- 2.2 Fréchet derivatives.- 2.3 Chain rule.- 2.4 Inverse function theorem.- 2.5 Implicit functions.- 2.6 Functional dependence.- 2.7 Higher derivatives.- 3. Maxima and Minima.- 3.1 Extrema and stationary points.- 3.2 Constrained minima and Lagrange multipliers.- 3.3 Discriminating constrained stationary points.- 3.4 Inequality constraints.- 3.5 Discriminating maxima and minima with inequality constraints 62 Further reading.- 4. Integrating Functions of Several Variables.- 4.1 Basic ideas of integration.- 4.2 Double integrals.- 4.3 Length, area and volume.- 4.4 Integrals over curves and surfaces.- 4.5 Differential forms.- 4.6 Stokes’s theorem.- Further reading.- Appendices.- A. Background required in linear algebra and elementary calculus.- B. Compact sets, continuous functions and partitions of unity.- C. Answers to selected exercises.- Index (including table of some special symbols).