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Download e-book for kindle: Advanced calculus of several variables by Edwards, Charles Henry

By Edwards, Charles Henry

ISBN-10: 0486683362

ISBN-13: 9780486683362

Modern conceptual therapy of multivariable calculus, emphasizing the interaction of geometry and research through linear algebra and the approximation of nonlinear mappings by means of linear ones. whilst, considerable recognition is paid to the classical functions and computational equipment. countless numbers of examples, difficulties and figures. 1973 edition.

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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.

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Advanced calculus of several variables by Edwards, Charles Henry


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