The Fifth Edition of this leading text offers substantial training in vectors and matrices, vector analysis, and partial differential equations. Vectors are introduced at the outset and serve at many points to indicate geometrical and physical significance of mathematical relations. Numerical methods are touched upon at various points, because of their practical value and the insights they give about theory.
- A chapter on ordinary differential equations has been restored as Chapter 9.
- Numerical methods are introduced throughout, both because of their practical value and because of the insights they give into the theory.
- A sound level of rigor is maintained throughout. Definitions are clearly labeled and all-important results are formulated as theorems.
- A few of the finer points of real variable theory are treated at the ends of Chapters 2, 4, and 6. A large number of problems (with answers) are distributed throughout the text. These include simple exercises as well as complex ones that stimulate critical reading.
- Generous references to the literature are given, and each chapter concludes with a list of books for supplementary reading. Starred sections are less essential in a first course.
Table of Contents
1. Vectors and Matrices.
Vectors in Space.
Lines and Planes.
Simultaneous Linear Equations.
Addition of Matrices
Scalar Times Matrix.
Multiplication of Matrices.
Inverse of a Square Matrix.
*Eigenvalues of a Square Matrix.
Analytic Geometry and Vectors n-Dimensional Space.
*Axioms for Vn.
Rank of a Matrix.
*Other Vector Spaces. 2. Differential Calculus of Functions of Several Variables.
Functions of Several Variables.
Domains and Regions.
Functional Notation Level Curves and Level Surfaces.
Limits and Continuity.
Total Differential Fundamental Lemma.
Differential of Functions of n Variables
The Jacobian Matrix.
Derivatives and Differentials of Composite Functions.
The General Chain Rule.
*Proof of a Case of the Implicit Function Theorem.
The Directional Derivative.
Partial Derivatives of Higher Order.
Higher Derivatives of Composite Functions.
The Laplacian in Polar, Cylindrical, and Spherical Coordinates.
Higher Derivatives of Implicit Functions.
Mixima and Minima of Functions of Several Variables.
*Extrema for Functions with Side Conditions
*Maxima and Minima of Quadratic Forms on the Unit Sphere.
*Real Variable Theory
Theorem on Maximum and Minimum. 3. Vector Differential Calculus.
Vector Fields and Scalar Fields.
The Gradient Field.
The Divergence of a Vector Field.
The Curl of a Vector Field.
*Curvilinear Coordinates in Space.
*Vector Operations in Orthogonal Curvilinear Coordinates.
*Tensors on a Surface or Hypersurface.
Exterior Product. 4. Integral Calculus of Functions of Several Variables.
The Definite Integral.
Numerical Evaluation of Indefinite Integrals.
Triple Integrals and Multiple Integrals in General.
Integrals of Vector Functions.
Change of Variables in Integrals.
Arc Length and Surface Area.
Improper Multiple Integrals.
Integrals depending on a Parameter—Leibnitz's Rule.
Existence of the Riemann Integral.
*Theory of Double Integrals. 5. Vector Integral Calculus.
Line Integrals in the Plane.
Integrals with Respect to Arc Length.
Basic Properties of Line Integrals.
Line Integrals as Integrals of Vectors.
Independence of Path.
Simply Connected Domains.
Extension of Results to Multiply Connected Domains.
Three-Dimensional Theory and Applications.
Line Integrals in Space.
Surfaces in Space.
The Divergence Theorem.
Integrals Independent of Path.
Irrotational and Solenoidal Fields.
*Change of Variables in a Multiple Integral.
*Potential Theory in the Plane.
*Green's Third Identity.
*Potential Theory in Space.
*Change of Variables in an m-Form and General Stokes's Theorem.
*Tensor Aspects of Differential Forms.
*Tensors and Differential Forms without Coordinates. 6. Infinite Series.
Upper and Lower Limits.
Further Properties of Sequences.
Tests for Convergence and Divergence.
Examples and Applications of Tests for Convergence and Divergence.
*Extended Ratio Test and Root Test.
*Computation with Series—Estimate of Error.
Operations on Series.
Sequences and Series of Functions.
Weierstrass M-Test for Uniform Convergence.
Properties of Uniformly Convergent Series and Sequences.
Taylor and MacLaurin Series.
Taylor's Formula with Remainder.
Further Operations on Power Series.
*Sequences and Series of Complex Numbers.
*Sequences and Series of Functions of Several Variables.
*Taylor's Formula for Functions of Several Variables.
*Improper Integrals Versus Infinite Series.
*Improper Integrals Depending on a Parameter—Uniform Convergence.
*Principal Value of Improper Integrals.
<F128>G-Function and B-Function.
*Convergence of Improper Multiple Integrals. 7. Fourier Series and Orthogonal Functions.
Convergence of Fourier Series.
Examples—Minimizing of Square Error.
Fourier Cosine Series.
Fourier Sine Series.
Remarks on Applications of Fourier Series.
Proof of Fundamental Theorem for Continuous, Periodic, and Piecewise Very Smooth Functions.
Proof of Fundamental Theorem.
*Fourier Series of Orthogonal Functions.
*Sufficient Conditions for Completeness.
*Integration and Differentiation of Fourier Series.
*Orthogonal Systems of Functions of Several Variables.
*Complex Form of Fourier Series.
*The Laplace Transform as a Special Case of the Fourier Transform.
General Functions. 8. Functions of a Complex Variable.
Complex-Valued Functions of a Real Variable.
Complex-Valued Functions of a Complex Variable.
Limits and Continuity.
Derivatives and Differentials.
The Functions log z, az, za, sin-1 z, cos-1 z.
Integrals of Analytic Functions.
Cauchy Integral Theorem.
Cauchy's Integral Formula.
Power Series as Analytic Functions.
Power Series Expansion of General Analytic Function.
Power Series in Positive and Negative Powers.
Isolated Singularities of an Analytic Function.
Zeros and Poles.
The Complex Number <F128>à.
Residue at Infinity.
Partial Fraction Expansion of Rational Functions.
Application of Residues to Evaluation of Real Integrals.
Definition of Conformal Mapping.
Examples of Conformal Mapping.
Applications of Conformal Mapping.
The Dirichlet Problem.
Dirichlet Problem for the Half-Plane.
Conformal Mapping in Hydrodynamics.
Applications of Conformal Mapping in the Theory of Elasticity.
Further Applications of Conformal Mapping.
General Formulas for One-to-One Mapping.
Schwarz-Christoffel Transformation. 9. Ordinary Differential Equations.
The Basic Problems.
Linear Differential Equations.
Systems of Differential Equations.
Linear Systems with Constant Coefficients.
A Class of Vibration Problems.
Solution of Differential Equations by Taylor Series.
The Existence and Uniqueness Theorem. 10. Partial Differential Equations.
Review of Equation for Forced Vibrations of a Spring.
Case of Two Particles.
Case of n Particles.
Fundamental Partial Differential Equation.
Classification of Partial Differential Equations.
The Wave Equation in One Dimension.
Properties of Solutions of the Wave Equation.
The One-Dimensional Heat Equation.
Properties of Solutions of the Heat Equation.
Equilibrium and Approach to Equilibrium.
Equations with Variable Coefficients.
Equations in Two and Three Dimensions.
Separation of Variables.
Partial Differential Equations and Integral Equations.