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AuthorCraven, B. D. author
TitleFunctions of several variables [electronic resource] / by B. D. Craven
ImprintDordrecht : Springer Netherlands, 1981
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Descript VIII, 138 p. online resource


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 (exยญ 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


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 Frรฉchet 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 Stokesโ{128}{153}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)

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