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Vector calculus, linear algebra, and differential forms : a unified approach

Author: John H Hubbard; Barbara Burke Hubbard
Publisher: Ithaca, NY : Matrix Editions, 2009.
Edition/Format:   Print book : English : 4th edView all editions and formats
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Document Type: Book
All Authors / Contributors: John H Hubbard; Barbara Burke Hubbard
ISBN: 9780971576650 0971576653 9780971576674 097157667X
OCLC Number: 319247418
Notes: Title on manual: Student solution manual to accompany the 4th edition of vector calculus, linear algebra, and differential forms.
Description: xiii, 818 pages : illustrations ; 26 cm + Student solution manual
Contents: CHAPTER 0 Preliminaries --
Section 0.0 Introduction --
Section 0.1 Reading mathematics --
Section 0.2 Quantifiers and negation --
Section 0.3 Set theory --
Section 0.4 Functions --
Section 0.5 Real numbers --
Section 0.6 Infinite sets --
Section 0.7 Complex numbers --
CHAPTER 1 Vectors, matrices, and derivatives --
Section 1.0 Introduction --
Section 1.1 Introducing the actors: points and vectors --
Section 1.2 Introducing the actors: matrices --
Section 1.3 Matrix multiplication as a linear transformation --
Section 1.4 The geometry of ${Bbb R}^n$ --
Section 1.5 Limits and continuity --
Section 1.6 Four big theorems --
Section 1.7 Derivatives in several variables as linear transformations --
Section 1.8 Rules for computing derivatives --
Section 1.9 The mean value theorem and criteria for differentiability --
Section 1.1 0 Review exercises for chapter 1 --
CHAPTER 2 Solving equations Section 2.0 Introduction --
Section 2.1 The main algorithm: row reduction --
Section 2.2 Solving equations with row reduction --
Section 2.3 Matrix inverses and elementary matrices --
Section 2.4 Linear combinations, span, and linear independence --
Section 2.5 Kernels, images, and the dimension formula --
Section 2.6 Abstract vector spaces --
Section 2.7 Eigenvectors and eigenvalues --
Section 2.8 Newton's method --
Section 2.9 Superconvergence --
Section 2.10 The inverse and implicit function theorems --
Section 2.11 Review exercises for chapter 2 --
CHAPTER 3 Manifolds, Taylor polynomials, quadratic forms, and curvature --
Section 3.0 Introduction --
Section 3.1 Manifolds --
Section 3.2 Tangent spaces --
Section 3.3 Taylor polynomials in several variables --
Section 3.4 Rules for computing Taylor polynomials --
Section 3.5 Quadratic forms --
Section 3.6 Classifying critical points of functions --
Section 3.7 Constrained critical points and Lagrange multipliers --
Section 3.8 Geometry of curves and surfaces --
Section 3.9 Review exercises for chapter 3 --
CHAPTER 4 Integration Section 4.0 Introduction --
Section 4.1 Defining the integral --
Section 4.2 Probability and centers of gravity --
Section 4.3 What functions can be integrated? --
Section 4.4 Measure zero --
Section 4.5 Fubini's theorem and iterated integrals --
Section 4.6 Numerical methods of integration --
Section 4.7 Other pavings --
Section 4.8 Determinants --
Section 4.9 Volumes and determinants --
Section 4.10 The change of variables formula --
Section 4.11 Lebesgue integrals --
Section 4.12 Review exercises for chapter 4 --
CHAPTER 5 Volumes of manifolds Section 5.0 Introduction --
Section 5.1 Parallelograms and their volumes --
Section 5.2 Parametrizations --
Section 5.3 Computing volumes of manifolds --
Section 5.4 Integration and curvature --
Section 5.5 Fractals and fractional dimension --
Section 5.6 Review exercises for chapter 5 --
CHAPTER 6 Forms and vector calculus --
Section 6.0 Introduction --
Section 6.1 Forms --
Section 6.2 Integrating form fields over parametrized domains --
Section 6.3 Orientation of manifolds --
Section 6.4 Integrating forms over oriented manifolds --
Section 6.5 Forms in the language of vector calculus --
Section 6.6 Boundary orientation --
Section 6.7 The exterior derivative --
Section 6.8 Grad, curl, div, and all that --
Section 6.9 Electromagnetism --
Section 6.10 The generalized Stokes's theorem --
Section 6.11 The integral theorems of vector calculus --
Section 6.12 Potentials --
Section 6.13 Review exercises for chapter 6 --
APPENDIX: Analysis A.0 --
A.1 Arithmetic of real numbers --
A.2 Cubic and quartic equations --
A.3 Two results in topology: nested compact sets and Heine-Borel --
A.4 Proof of the chain rule --
A.5 Proof of Kantorovich's theorem --
A.6 Proof of lemma 2.9.5 (superconvergence) --
A.7 Proof of differentiability of the inverse function --
A.8 Proof of the implicit function theorem --
A.9 Proof of theorem 3.3.9: equality of crossed partials --
A.10 Functions with many vanishing partial derivatives --
A.11 Proving rules for Taylor polynomials; big O and little o --
A.12 Taylor's theorem with remainder --
A.13 Proof of theorem 3.5.3 (completing squares) --
A.14 Geometry of curves and surfaces: proofs --
A.15 Stirling's formula and proof of the central limit theorem --
A.16 Proof of Fubini's theorem --
A.17 Justifying the use of other pavings --
A.18 Results concerning the determinant --
A.19 Change of variables formula: a rigorous proof --
A.20 Justifying volume 0 --
A.21 Lebesgue measure and proofs for Lebesgue integrals --
A.22 Justifying the change of parametrization --
A.23 Computing the exterior derivative --
A.24 The pullback --
A.25 Proof of Stokes's theorem.
Other Titles: Student solution manual to accompany the 4th edition of vector calculus, linear algebra, and differential forms.
Responsibility: John Hamal Hubbard, Barbara Burke Hubbard.

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