# Multivariable Calculus, Linear Algebra, and Differential Equations

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Multivariable Calculus, Linear Algebra, and Differential Equations, Second Edition contains a comprehensive coverage of the study of advanced calculus, linear algebra, and differential equations for sophomore college students. The text includes a large number of examples, exercises, cases, and applications for students to learn calculus well. Also included is the history and development of calculus. The book is divided into five parts. The first part includes multivariable calculus material. The second part is an introduction to linear algebra. The third part of the book combines techniques from calculus and linear algebra and contains discussions of some of the most elegant results in calculus including Taylor's theorem in "n" variables, the multivariable mean value theorem, and the implicit function theorem. The fourth section contains detailed discussions of first-order and linear second-order equations. Also included are optional discussions of electric circuits and vibratory motion. The final section discusses Taylor's theorem, sequences, and series. The book is intended for sophomore college students of advanced calculus.

## Table of Contents

Preface

1 Vectors In The Plane

1.1 Vectors and Vector Operations

1.2 The Dot Product

1.3 Some Applications of Vectors (Optional)

Review Exercises for Chapter One

2 Vector Functions, Vector Differentiation, And Parametric Equations In R2

2.1 Vector Functions and Parametric Equations

2.2 The Equation of the Tangent Line to a Parametric Curve

2.3 The Differentiation and Integration of a Vector Function

2.4 Some Differentiation Formulas

2.5 Arc Length Revisited

2.6 Arc Length as a Parameter

2.7 Velocity, Acceleration, Force, and Momentum

2.8 Curvature and the Acceleration Vector (Optional)

Review Exercises for Chapter Two

3 Vectors In Space

3.1 The Rectangular Coordinate System in Space

3.2 Vectors in R3

3.3 Lines in R3

3.4 The Cross Product of Two Vectors

3.5 Planes

3.6 Quadric Surfaces

3.7 The Space Rn and the Scalar Product

3.8 Vector Functions and Parametric Equations in R3

3.9 Cylindrical and Spherical Coordinates

Review Exercises for Chapter Three

4 Differentiation Of Functions Of Two Or More Variables

4.1 Functions of Two or More Variables

4.2 Limits and Continuity

4.3 Partial Derivatives

4.4 Higher-Order Partial Derivatives

4.5 Differentiability and the Gradient

4.6 The Chain Rules

4.7 Tangent Planes, Normal Lines, and Gradients

4.8 Directional Derivatives and the Gradient

4.9 Conservative Vector Fields and the Gradient (Optional)

4.10 The Total Differential and Approximation

4.11 Exact Vector Fields or How to Obtain a Function from Its Gradient

4.12 Maxima and Minima for a Function of Two Variables

4.13 Constrained Maxima and Minima—Lagrange Multipliers

Review Exercises for Chapter Four

5 Multiple Integration

5.1 Volume Under a Surface and the Double Integral

5.2 The Calculation of Double Integrals

5.3 Density, Mass, and Center of Mass (Optional)

5.4 Double Integrals in Polar Coordinates

5.5 Surface Area

5.6 The Triple Integral

5.7 The Triple Integral in Cylindrical and Spherical Coordinates

Review Exercises for Chapter Five

6 Introduction To Vector Analysis

6.1 Vector Fields

6.2 Work, Line Integrals in the Plane, and Independence of Path

6.3 Green's Theorem in the Plane

6.4 Line Integrals in Space

6.5 Surface Integrals

6.6 Divergence and Curl of a Vector Field in R3

6.7 Stokes's Theorem

6.8 The Divergence Theorem

6.9 Changing Variables in Multiple Integrals and the Jacobian

Review Exercises for Chapter Six

7 Matrices And Linear Systems Of Equations

7.1 Matrices

7.2 Matrix Products

7.3 Linear Systems of Equations

7.4 Matrices and Linear Systems of Equations

7.5 The Inverse of a Square Matrix

7.6 The Transpose of a Matrix

Review Exercises for Chapter Seven

8 Determinants

8.1 Definitions

8.2 Properties of Determinants

8.3 Determinants and Inverses

8.4 Cramer's Rule (Optional)

Review Exercises for Chapter Eight

9 Vector Spaces And Linear Transformations

9.1 Vector Spaces

9.2 Subspaces

9.3 Linear Independence, Linear Combination and Span

9.4 Basis and Dimension

9.5 Change of Basis (Optional)

9.6 Linear Transformations

9.7 Properties of Linear Transformations: Range and Kernel

9.8 The Rank and Nullity of a Matrix

9.9 The Matrix Representation of a Linear Transformation

9.10 Eigenvalues and Eigenvectors

9.11 If Time Permits: A Model of Population Growth

9.12 Similar Matrices and Diagonalization

Review Exercises for Chapter Nine

10 Calculus In Rn

10.1 Taylor's Theorem in n Variables

10.2 Inverse Functions and the Implicit Function Theorem: I

10.3 Functions from Rn to Rm

10.4 Derivatives and the Jacobian Matrix

10.5 Inverse Functions and the Implicit Function Theorem: II

Review Exercises for Chapter Ten

11 Ordinary Differential Equations

11.1 Introduction

11.2 First-Order Equations—Separation of Variables

11.3 Exact Equations (Optional)

11.4 First-Order Linear Equations

11.5 Simple Electric Circuits (Optional)

11.6 Second-Order Linear Differential Equations: Theory

11.7 Using One Solution to Find Another

11.8 Homogeneous Equations with Constant Coefficients: Real Roots

11.9 Homogeneous Equations with Constant Coefficients: Complex Roots

11.10 Nonhomogeneous Equations: The Method of Undetermined Coefficients

11.11 Nonhomogeneous Equations: Variation of Constants

11.12 Euler Equations

11.13 Vibratory Motion (Optional)

11.14 More On Electric Circuits (Optional)

11.15 Higher-Order Linear Differential Equations

11.16 Numerical Solution of Differential Equations: Euler's Methods

Review Exercises for Chapter Eleven

12 Matrices And Systems Of Differential Equations

12.1 The Method of Elimination for Linear Systems with Constant Coefficients

12.2 Linear Systems: Theory

12.3 The Solution of Homogeneous Linear Systems with Constant Coefficients: The Method of Determinants

12.4 Matrices and Systems of Linear First-Order Equations

12.5 Fundamental Sets and Fundamental Matrix Solutions of a Homogeneous System of Differential Equations

12.6 The Computation of the Principal Matrix Solution to a Homogeneous System of Equations

12.7 Nonhomogeneous Systems

12.8 An Application of Nonhomogeneous Systems: Forced Oscillations (Optional)

Review Exercises for Chapter Twelve

13 Taylor Polynomials, Sequences, And Series

13.1 Taylor's Theorem and Taylor Polynomials

13.2 A Proof of Taylor's Theorem, Estimates on the Remainder Term, and a Uniqueness Theorem (Optional)

13.3 Approximation Using Taylor Polynomials

13.4 Sequences of Real Numbers

13.5 Bounded and Monotonic Sequences

13.6 Geometric Series

13.7 Infinite Series

13.8 Series with Nonnegative Terms I: Two Comparison Tests and the Integral Test

13.9 Series with Nonnegative Terms II: The Ratio and Root Tests

13.10 Absolute and Conditional Convergence: Alternating Series

13.11 Power Series

13.12 Differentiation and Integration of Power Series

13.13 Taylor and Maclaurin Series

13.14 Using Power Series to Solve Ordinary Differential Equations (Optional)

Review Exercises for Chapter Thirteen

Appendix 1 Mathematical Induction

Appendix 2 The Binomial Theorem

Appendix 3 Complex Numbers

Appendix 4 Proof Of The Basic Theorem About Determinants

Appendix 5 Existence And Uniqueness For First Order Initial Value Problems

Table Of Integrals

Answers to Odd-Numbered Problems and Review Exercises

Index

## Product details

- No. of pages: 992
- Language: English
- Copyright: © Academic Press 1986
- Published: January 1, 1986
- Imprint: Academic Press
- eBook ISBN: 9781483218038

## About the Author

### Stanley I. Grossman

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