Multivariable and Vector Calculus. An Introduction

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Cover Half-Title Title Copyright Dedication Contents Preface Acknowledgments Chapter 1: Vectors and Parametric Curves 1.1 Points and Vectors on the Plane Exercises 1.1 1.2 Scalar Product on the Plane Exercises 1.2 1.3 Linear Independence Exercises 1.3 1.4 Geometric Transformations in Two Dimensions Exercises 1.4 1.5 Determinants in Two Dimensions Exercises 1.5 1.6 Parametric Curves on the Plane Exercises 1.6 1.7 Vectors in Space Exercises 1.7 1.8 Cross Product Exercises 1.8 1.9 Matrices in Three Dimensions Exercises 1.9 1.10 Determinants in Three Dimensions Exercises 1.10 1.11 Some Solid Geometry Exercises 1.11 1.12 Cavalieri and the Pappus–Guldin Rules Exercises 1.12 1.13 Dihedral Angles and Platonic Solids Exercises 1.13 1.14 Spherical Trigonometry Exercises 1.14 1.15 Canonical Surfaces Exercises 1.15 1.16 Parametric Curves in Space Exercises 1.16 1.17 Multidimensional Vectors Exercises 1.17 Chapter 2: Differentiation 2.1 Some Topology Exercises 2.1 2.2 Multivariable Functions Exercises 2.2 2.3 Limits and Continuity Exercises 2.3 2.4 Definition of the Derivative Exercises 2.4 2.5 The Jacobi Matrix Exercises 2.5 2.6 Gradients and Directional Derivatives Exercises 2.6 2.7 Levi-Civitta and Einste Exercises 2.7 2.8 Extrema Exercises 2.8 2.9 Lagrange Multipliers Exercises 2.9 Chapter 3: Integration 3.1 Differential Forms Exercises 3.1 3.2 Zero-Manifolds Exercises 3.2 3.3 One Manifold Exercises 3.3 3.4 Closed and Exact Forms Exercises 3.4 3.5 Two-Manifolds Exercises 3.5 3.6 Change of Variables in Double Integrals Exercises 3.6 3.7 Change to Polar Coordinates Exercises 3.7 3.8 Three-Manifolds Exercises 3.8 3.9 Change of Variables in Triple Integrals Exercises 3.9 3.10 Surface Integrals Exercises 3.10 3.11 Green’s, Stokes’, and Gauss’ Theorems Exercises 3.11 Appendix A: Maple A.1 Getting Started and Windows of Maple A.2 Arithmetic A.3 Symbolic Computation A.4 Assignments A.5 Working with Output A.6 Solving Equations A.7 Plots with Maple A.8 Limits and Derivatives A.9 Integration A.10 Matrix Appendix B: Matlab B.1 Getting Started and Windows of MATLAB B.1.1 Using MATLAB in Calculations B.2 Plotting B.2.1 Two-dimensional Plotting B.2.2 Three-Dimensional Plotting B.3 Programming in MATLAB B.3.1 For Loops B.3.2 While Loops B.3.3 If, Else, and Elseif 3.3.4 Switch B.4 Symbolic Computation B.4.1 Simplifying Symbolic Expressions B.4.2 Differentiating Symbolic Expressions B.4.3 Integrating Symbolic Expressions B.4.4 Limits Symbolic Expressions B.4.5 Taylor Series Symbolic Expressions B.4.6 Sums Symbolic Expressions B.4.7 Solving Equations as Symbolic Expressions Appendix C: Answers to Odd-Numbered Exercises Chapter 1 Chapter 2 Chapter 3 Appendix D: Formulas D.1 Trigonometric Identities D.2 Hyperbolic Functions D.3 Table of Derivatives D.4 Table of Integrals D.5 Summations (Series) D.5.1 Finite Element of Terms D.5.2 Infinite Element of Terms D.6 Logarithmic Identities D.7 Exponential Identities D.8 Approximations for Small Quantities D.9 Vectors D.9.1 Vector Derivatives D.9.2 Vector Identity D.9.3 Fundamental Theorems Bibliography Index