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1 T H I R D E D I T I O N MULTIVARIABLE CALCULUS, LINEAR ALGEBRA, AND DIFFERENTIAL EQUATIONS STANLEY I. GROSSMAN University of Montana and University College London SAUNDERS COLLEGE PUBLISHING HARCOURT BRACE COLLEGE PUBLISHERS Fort Worth Philadelphia San Diego New York Orlando Austin San Antonio Toronto Montreal London Sydney Tokyo

2 TABLE OF CONTENTS MULTIVARIABLE CALCULUS VECTORS IN THE PLANE AND IN SPACE, 3 Vectors and Vector Operations, 4 The Dot Product, 14 The Rectangular Coordinate System in Space, 24 Vectors in U\ 28 Lines in IR 3, 36 The Cross Product of Two Vectors, 42 Planes, 52 Quadric Surfaces, 59 Cylindrical and Spherical Coordinates, 66 The Space W and the Scalar Product (Optional), 72 Summary Outline, 80 Review Exercises, 83 2 VECTOR FUNCTIONS AND PARAMETRIC EQUATIONS, Vector Functions and Parametric Equations, The Equation of the Tangent Line to a Plane Curve and Smoothness, The Differentiation and Integration of a Vector Function, Some Differentiation Formulas, Arc Length Revisited, Curvature and the Acceleration Vector (Optional), 120 Summary Outline, 131 Review Exercises, 133 Computer Exercises, DIFFERENTIATION OF FUNCTIONS OF TWO OR MORE VARIABLES, 1 35 Functions of Two or More Variables, 136 Limits and Continuity, 147 Partial Derivatives, 162 Higher-Order Partial Derivatives, 170 Differentiability and the Gradient, 178 The Chain Rule, 186 Tangent Planes, Normal Lines, and Gradients, 197 Directional Derivatives and the Gradient, 201 The Total Differential and Approximation, 210 Maxima and Minima for a Function of Two Variables, 212 Constrained Maxima and Minima Lagrange Multipliers, 222 Newton's Method for Functions of Two Variables (Optional), 232 Summary Outline, 236 Review Exercises, 240 Computer Exercises, 242 MULTIPLE INTEGRATION, 243 Volume Under a Surface and the Double Integral, 243 The Calculation of Double Integrals, 253 Density, Mass, and Center of Mass (Optional), 265 Double Integrals in Polar Coordinates, 269 The Triple Integral, 275 The Triple Integral in Cylindrical and Spherical Coordinates, 282 Summary Outline, 287 Review Exercises, 289 Computer Exercises, 290 XIII

3 XIV TABLE OF CONTENTS 5 INTRODUCTION TO VECTOR ANALYSIS, Vector Fields, Work and Line Integrals, Exact Vector Fields and Independence of Path, Green's Theorem in the Plane, The Parametric Representation of a Surface and Surface Area, Surface Integrals, Divergence and Curl, Stokes's Theorem, The Divergence Theorem, Changing Variables in Multiple Integrals and the Jacobian, 356 Summary Outline, 364 Review Exercises, 367 LINEAR ALGEBRA SYSTEMS OF LINEAR EQUATIONS AND MATRICES, 371 Introduction, 371 Two Linear Equations in Two Unknowns, 372 m Equations in n Unknowns: Gauss-Jordan and Gaussian Elimination, 376 Homogeneous Systems of Equations, 390 Matrices, 393 Matrix Products, 400 Matrices and Linear Systems of Equations, 410 The Inverse of a Square Matrix, 415 The Transpose of a Matrix, 432 Elementary Matrices and Matrix Inverses, 436 Summary Outline, 444 Review Exercises, 446 DETERMINANTS, 448 Definitions, 448 Properties of Determinants, 457 Determinants and Inverses, 473 Cramer's Rule (Optional), 479 Summary Outline, 483 Review Exercises, VECTOR SPACES AND LINEAR TRANSFORMATIONS, Vector Spaces, Subspaces, Linear Combination and Span, Linear Independence, Basis and Dimension, The Rank, Nullity, Row Space, and Column Space of a Matrix, Orthonormal Bases and Projections in W Least Squares Approximation, Linear Transformations, Properties of Linear Transformations: Range and Kernel, The Matrix Representation of a Linear Transformation, Isomorphisms, Eigenvalues and Eigenvectors, A Model of Population Growth (Optional), Similar Matrices and Diagonalization, Symmetric Matrices and Orthogonal Diagonalization, Quadratic Forms and Conic Sections, 615 Summary Outline, 624 Review Exercises, 628

4 TABLE OF CONTENTS XV PART III INTRODUCTION TO INTERMEDIATE CALCULUS CALCULUS IN U n, 632 Taylor's Theorem in n Variables, 632 Inverse Functions and the Implicit Function Theorem: I, 642 Functions from W to W, Derivatives and the Jacobian Matrix, Inverse Functions and the Implicit Function Theorem: II, 663 Summary Outline, 674 Review Exercises, 676 PART IV DIFFERENTIAL EQUATIONS 10 ORDINARY DIFFERENTIAL EQUATIONS, Introduction, Review of the Differential Equation of Exponential Growth, First-Order Equations Separation of Variables, Linear First Order Differential Equations, Exact Differential Equations (Optional), Simple Electric Circuits, Theory of Linear Differential Equations, Using One Solution to Find Another: Reduction of Order, Homogeneous Equations with Constant Coefficients: Real Roots, Homogeneous Equations with Constant Coefficients: Complex Roots, Nonhomogeneous Equations I: Variation of Parameters, Nonhomogeneous Equations II: The Method of Undetermined Coefficients, Euler Equations, Vibratory Motion (Optional), More on Electric Circuits (Optional), Higher-Order Linear Differential Equations (Optional), ' Numerical Solution of Differential Equations: Euler's Methods, 797 Summary Outline, 803 Review Exercises, SYSTEMS OF DIFFERENTIAL EQUATIONS, 8O The Method of Elimination for Linear Systems with Constant Coefficients, Linear Systems: Theory, The Solution of Homogeneous Linear Systems with Constant Coefficients: The Method of Determinants, Electric Circuits with Several Loops (Optional), Mechanical Systems, A Model for Epidemics (Optional), Matrices and Systems of Linear First-Order Equations Fundamental Sets and Fundamental Matrix Solutions of a Homogeneous System of Differential Equations, The Computation of the Principal Matrix Solution to a Homogeneous System of Equations with Constant Coefficients, Nonhomogeneous Systems, An Application of Nonhomogeneous Systems: Forced Oscillations (Optional), 880

5 XVI TABLE OF CONTENTS Computing e At : An Application of the Cayley- Hamilton Theorem (Optional), 884 Summary Outline, 893 Review Exercises, 897 PART V SEQUENCES AND SERIES 12 TAYLOR POLYNOMIALS, SEQUENCES, AND SERIES, Taylor's Theorem and Taylor Polynomials, Approximation Using Taylor Polynomials, Sequences of Real Numbers, Bounded and Monotonic Sequences, Geometric Series, Infinite Series, Series with Nonnegative Terms I: Two Comparison Tests and the Integral Test, Series with Nonnegative Terms II: The Ratio and Root Tests, Absolute and Conditional Convergence: Alternating Series, Power Series, Differentiation and Integration of Power Series, Taylor and Maclaurin Series, Using Power Series to Solve Ordinary Differential Equations (Optional), 994 Summary Outline, 1002 Review Exercises, 1006 Computer Exercises, 1008 APPENDIX 1 MATHEMATICAL INDUCTION, A-1 APPENDIX 2 THE BINOMIAL THEOREM, A-6 APPENDIX 3 COMPLEX NUMBERS, A-14 APPENDIX 4 PROOF OF THE BASIC THEOREM ABOUT DETERMINANTS, A-21 APPENDIX 5 EXISTENCE AND UNIQUENESS FOR FIRST- ORDER INITIAL-VALUE PROBLEMS, A-24 APPENDIX 6 THE FOUNDATIONS OF VECTOR SPACE THEORY: THE EXISTENCE OF A BASIS, A-37 TABLE OF INTEGRALS, A-43 ANSWERS TO ODD-NUMBERED PROBLEMS AND REVIEW EXERCISES, A-51 INDEX, 1-1

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