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Multivariable Calculus, Linear Algebra, and Differential Equations
- 2nd Edition - May 10, 2014
- Author: Stanley I. Grossman
- Language: English
- eBook ISBN:9 7 8 - 1 - 4 8 3 2 - 1 8 0 3 - 8
Multivariable Calculus, Linear Algebra, and Differential Equations, Second Edition contains a comprehensive coverage of the study of advanced calculus, linear algebra, and… Read more
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Request a sales quoteMultivariable 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.
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
- No. of pages: 992
- Language: English
- Edition: 2
- Published: May 10, 2014
- Imprint: Academic Press
- eBook ISBN: 9781483218038
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