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Matrix Editions: Serious mathematics, written with the reader in mind.

 Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach ISBN 9780971576650 \$84, hardcover, 818 pages, smythe-sewn binding   4th edition September 2009 In chapters 0-6 we have put in red sections where the page numbers have been changed, compared to the 3rd edition. Contents   Preface           xi     Chapter 0     Preliminaries     0.0     Introduction      1   0.1     Reading mathematics  1   0.2     Quantifiers and negation   4   0.3     Set theory   6   0.4     Functions   9   0.5     Real numbers  17   0.6     Infinite sets  22   0.7     Complex numbers   25      Chapter 1   Vectors, matrices, and derivatives           1.0   Introduction   32   1.1   Introducing the actors:  points and vectors  33   1.2   Introducing the actors:  matrices       42   1.3   Matrix multiplication as a linear transformation  56           1.4   The geometry of  Rn     67           1.5   Limits and continuity    84           1.6   Four big theorems   106           1.7   Derivatives in several variables as linear transformations  120           1.8   Rules for computing derivatives    140           1.9   The mean value theorem and criteria for differentiability   148           1.10  Review exercises for chapter 1    155    Chapter 2    Solving equations                  2.0   Introduction  161           2.1   The main algorithm:  row reduction   162           2.2   Solving equations with  row reduction   168           2.3   Matrix inverses and elementary matrices   177           2.4   Linear combinations, span, and linear independence  182           2.5   Kernels, images, and the dimension formula   195           2.6   Abstract vector spaces    211           2.7   Eigenvectors and eigenvalues   222           2.8   Newton's method   232            2.9   Superconvergence   252            2.10 The inverse and implicit function theorems   259           2.11  Review exercises for chapter 2    278    Chapter 3  Manifolds, Taylor polynomials, quadratic forms,  and curvature                3.0   Introduction  284            3.1   Manifolds  285            3.2   Tangent spaces   306           3.3   Taylor polynomials in several variables   314           3.4   Rules for computing Taylor polynomials   326           3.5   Quadratic forms   334           3.6   Classifying critical points of functions   344            3.7   Constrained critical points and Lagrange multipliers  351           3.8   Geometry of curves and surfaces             3.9   Review exercises for chapter 3   389      Chapter 4  Integration                4.0    Introduction   393             4.1    Defining the integral   394             4.2    Probability and centers of gravity  409             4.3    What functions can be integrated?   423             4.4    Measure zero      430             4.5    Fubini's theorem and iterated integrals   438             4.6    Numerical methods of integration   450             4.7    Other pavings   461             4.8    Determinants   463             4.9   Volumes and determinants   478             4.10  The change of variables formula   485             4.11  Lebesgue integrals   497             4.12  Review exercises for chapter 4      516    Chapter 5  Volumes of manifolds                5.0    Introduction  520             5.1    Parallelograms and their volumes 521             5.2    Parametrizations 524             5.3    Computing  volumes of manifolds 532             5.4    Integration and curvature 544             5.5    Fractals and fractional dimension  554             5.6    Review exercises for chapter 5   556    Chapter 6   Forms and vector calculus            6.0    Introduction  558             6.1    Forms on Rn  559              6.2    Integrating form fields over parametrized domains  572             6.3    Orientation of manifolds   576             6.4    Integrating forms over oriented manifolds  584             6.5    Forms in the language of vector calculus  594             6.6    Boundary orientation   606             6.7    The exterior derivative  620             6.8    Grad, curl, div, and all that 627               6.9    The generalized Stokes's theorem  635            6.10  The integral theorems of vector calculus   643            6.12  Electromagnetism  667            6.13  Potentials    667            6.13  Review exercises for chapter 6   668  Appendix:  Analysis                       A.0    Introduction  683            A.1    Arithmetic of real numbers  683            A.2    Cubic and quartic equations  687            A.3    Two   results in topology: nested compact sets and                           Heine-Borel  692            A.4    Proof of the chain rule    693            A.5    Proof of Kantorovich's theorem   696            A.6    Proof of lemma 2.9.5  (superconvergence)  702            A.7    Proof of  differentiability of the inverse function  704            A.8    Proof of the implicit function  theorem  706            A. 9   Proving equality of crossed partials  711            A.10  Functions with many vanishing partial derivatives 712            A.11  Proving rules for Taylor polynomials; big  O  and                           little o  715            A.12  Taylor's theorem with remainder 720            A.13  Proving theorem 3.5.3  (completing squares)  725            A.14  Classifying constrained critical points  726 (new section)            A.15  Geometry of curves and surfaces: proofs   730            A.16  Stirling's formula and proof of  the central limit theorem   735            A.17  Proving Fubini's theorem  740            A.18  Justifying the use of other pavings  742            A.19  Results concerning the determinant  745            A.20  Change of variables formula: a rigorous proof   750            A.21  Justifying volume 0  756            A.22  Lebesgue measure and proofs for Lebesgue integrals  758            A.23  Justifying the change of parametrization 776            A.24  Computing the exterior derivative 781            A.25 The pullback   785            A.26  Proving Stokes's theorem 790     Bibliography   804     Photo credits   805     Index  607  To order (within the United States)