Vector Calculus and Differential Equations

Cover
Copyright
Preface
Contents
Chapter 1 Vectors
1.1 Elementary vector geometry of 3-space
1.2 Linear dependence and independence
1.3 The dot product
1.4 The vector (or cross) product
1.5 Triple products of vectors
1.6 The line and plane
1.7 Vector spaces
1.8 Vector spaces with inner products
Chapter 2 Vector Functions
2.1 Vector functions
2.2 Limits of vector functions
2.3 Derivatives of vector functions
2.4 Curves, directed curves, and arclength
2.5 Elementary kinematics
2.6 Introductory differential geometry
2.7 Curvature of arc
2.8 The Frenet-Serret formulas
Chapter 3 Differential Calculus of R^n
3.1 R^n-R^m functions
3.2 Limits and continuity for R^n-R^m functions
3.3 Surfaces
3.4 Partial derivatives
3.5 Higher-order partial derivatives
3.6 M.V.T. for partial derivatives
3.7 The gradient
3.8 Differentiability
3.9 The general chain rule for derivatives
3.10 Gradient fields
3.11 The direction derivative
3.12 Implicit differentiation
3.13 Taylor's theorem for R^n-R functions
3.14 Extrema for R^n-R functions
Chapter 4 Multiple Integrals
4.1 Area and volume
4.2 The double integral
4.3 Iterated integration
4.4 Double integrals in polar coordinates
4.5 The triple integral
4.6 Mass and moments
4.7 Cylindrical and spherical coordinates
4.8 The area of an explicit surface
Chapter 5 Infinite Series
5.1 Sequential limits
5.2 Convergence and divergence of series
5.3 Comparison tests for positive-termed series
5.4 Alternating series
5.5 Absolute and conditional convergence
5.6 The ratio and root tests
5.7 Power series
5.8 Differentiation and integration of power series
5.9 The binomial series
Chapter 6 Line Integrals
6.1 Line integrals with respect to arclength
6.2 Line integrals of vector functions
6.3 Green's theorem
6.4 Exactness and path-independence
Chapter 7 Advanced Concepts in Infinite Series
7.1 Rearrangement of series
7.2 Multiplication of series
7.3 The substitution theorem for power series
7.4 Inversion of power series
7.5 Pointwise and uniform convergence of function sequences
7.6 Termwise integration and differentiation
7.7 Series of functions
7.8 Complex series and elementary complex functions
Chapter 8 Improper Integrals and Integrals with Parameter
8.1 The definition of improper integral
8.2 Convergence tests for improper integrals
8.3 Proper integrals with parameter
8.4 Improper integrals with parameter
Chapter 9 Introductory Differential Equations
9.1 Differential equations
9.2 Families of curves and associated differential equations
9.3 Initial-value problems and existence-uniqueness theorems
9.4 Separable equations
9.5 Homogeneous equations
9.6 First-order linear equations
9.7 Exact equations
9.8 Orthogonal trajectories
9.9 Applications to elementary problems in motion
9.10 Picard's method of successive approximations
9.11 Proof of the existence-uniqueness theorem
Chapter 10 Linear Differential Equations
10.1 Existence and uniqueness
10.2 Second-order equations
10.3 Linear dependence and independence
10.4 Differential operators with constant coefficients
10.5 Non-homogeneous equations: undetermined coefficients
10.6 Variation of parameters
10.7 Reduction of order by known solutions
10.8 The Euler equation
10.9 Applications to vibrational systems
10.10 Applications to electrical circuits
10.11 Systems of differential equations
Chapter 11 The Laplace Transform
11.1 The definition of Laplace transform
11.2 Table of transforms
11.3 Algebraic properties
11.4 Inverse transforms
11.5 Transforms of derivatives
11.6 Solving differential equations by transforms
11.7 Systems of equations solved by Laplace transforms
11.8 Derivatives of Laplace transforms
11.9 Convolution theorem
Chapter 12 Series Solutions
12.1 Power series solutions
12.2 Power series solutions: successive differentiations
12.3 Solutions about a regular singular point
12.4 Bessel's equation
Chapter 13 Fourier Series
13.1 Trigonometric Fourier series
13.2 Convergence of Fourier series
13.3 Half-range Fourier series
13.4 Differentiation and integration of Fourier series
13.5 Fourier series with respect to orthogonal sets
13.6 Application to the vibrating string problem
Appendix A
Selected Answers
Index