By Edwards, Charles Henry

ISBN-10: 0486683362

ISBN-13: 9780486683362

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Com To My Parents CONTENTS Preface I Euclidean Space and Linear Mappings 1 The Vector Space n 2 Subspaces of n 3 Inner Products and Orthogonality 4 Linear Mappings and Matrices 5 The Kernel and Image of a Linear Mapping 6 Determinants 7 Limits and Continuity 8 Elementary Topology of n IIMultivariable Differential Calculus 1 Curves in m 2 Directional Derivatives and the Differential 3 The Chain Rule 4 Lagrange Multipliers and the Classification of Critical Points for Functions of Two Variables 5 Maxima and Minima, Manifolds, and Lagrange Multipliers 6 Taylor's Formula for Single-Variable Functions 7 Taylor's Formula in Several Variables 8 The Classification of Critical Points IIISuccessive Approximations and Implicit Functions 1 Newton's Method and Contraction Mappings 2 The Multivariable Mean Value Theorem 3 The Inverse and Implicit Mapping Theorems 4 Manifolds in n 5 Higher Derivatives IV Multiple Integrals 1 Area and the 1-Dimensional Integral 2 Volume and the n-Dimensional Integral 3 Step Functions and Riemann Sums 4 Iterated Integrals and Fubini's Theorem 5 Change of Variables 6 Improper Integrals and Absolutely Integrable Functions V Line and Surface Integrals; Differential Forms and Stokes' Theorem 1 Pathlength and Line Integrals 2 Green's Theorem 3 Multilinear Functions and the Area of a Parallelepiped 4 Surface Area 5 Differential Forms 6 Stokes' Theorem 7 The Classical Theorems of Vector Analysis 8 Closed and Exact Forms VI The Calculus of Variations 1 Normed Vector Spaces and Uniform Convergence 2 Continuous Linear Mappings and Differentials 3 The Simplest Variational Problem 4 The Isoperimetric Problem 5 Multiple Integral Problems Appendix: The Completeness of Suggested Reading Subject Index PREFACE This book has developed from junior–senior level advanced calculus courses that I have taught during the past several years.

Now let an m × n matrix A be given, and define a function f: n → m by f( x) = A x. Then f is linear, because by the distributivity of the scalar product of vectors, so f( x + y) = f( x) + f( y), and f(r x) = rf( x) similarly. The following theorem asserts not only that every mapping of the form f( x) = A x is linear, but conversely that every linear mapping from n to m is of this form. 1 The mapping f: n → m is linear if and only if there exists a matrix A such that f( x) = A x for all . Then A is that m × n matrix whose jth column is the column vector f( ej), where ej = (0, .

N constitute a basis for V*. 6 DETERMINANTS It is clear by now that a method is needed for deciding whether a given n-tuple of vectors a1, . . , an in n are linearly independent (and therefore constitute a basis for n). We discuss in this section the method of determinants. The determinant of an n × n matrix A is a real number denoted by det A or A. The student is no doubt familiar with the definition of the determinant of a 2 × 2 or 3 × 3 matrix. If A is 2 × 2, then For 3 × 3 matrices we have expansions by rows and columns.

### Advanced calculus of several variables by Edwards, Charles Henry

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