 # Advanced Calculus for Applications, 2nd edition

• Francis B. Hildebrand

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## Overview

The text provides advanced undergraduates with the necessary background in advanced calculus topics, providing the foundation for partial differential equations and analysis. Readers of this text should be well-prepared to study from graduate-level texts and publications of similar level.

KEY TOPICS: Ordinary Differential Equations; The Laplace Transform; Numerical Methods for Solving Ordinary Differential Equations; Series Solutions of Differential Equations: Special Functions; Boundary-Value Problems and Characteristic-Function Representations; Vector Analysis; Topics in Higher-Dimensional Calculus; Partial Differential Equations; Solutions of Partial Differential Equations of Mathematical Physics; Functions of a Complex Variable; Applications of Analytic Function Theory

1. Ordinary Differential Equations

1.1 Introduction

1.2 Linear Dependence

1.3 Complete Solutions of Linear Equations

1.4 The Linear Differential Equation of First Order

1.5 Linear Differential Equations with Constant Coefficients

1.6 The Equidimensional Linear Differential Equation

1.7 Properties of Linear Operators

1.8 Simultaneous Linear Differential Equations

1.9 particular Solutions by Variation of Parameters

1.10 Reduction of Order

1.11 Determination of Constants

1.12 Special Solvable Types of Nonlinear Equations

2. The Laplace Transform

2.1 An introductory Example

2.2 Definition and Existence of Laplace Transforms

2.3 Properties of Laplace Transforms

2.4 The Inverse Transform

2.5 The Convolution

2.6 Singularity Functions

2.7 Use of Table of Transforms

2.8 Applications to Linear Differential Equations with Constant Coefficients

2.9 The Gamma Function

3. Numerical Methods for Solving Ordinary Differential Equations

3.1 Introduction

3.2 Use of Taylor Series

3.5 The Runge-Kutta Method

3.6 Picard's Method

3.7 Extrapolation with Differences

4. Series Solutions of Differential Equations: Special Functions

4.1 Properties of Power Series

4.2 Illustrative Examples

4.3 Singular Points of Linear Second-Order Differential Equations

4.4 The Method of Frobenius

4.5 Treatment of Exceptional Cases

4.6 Example of an Exceptional Case

4.7 A Particular Class of Equations

4.8 Bessel Functions

4.9 Properties of Bessel Functions

4.10 Differential Equations Satisfied by Bessel Functions

4.11 Ber and Bei Functions

4.12 Legendre Functions

4.13 The Hypergeometric Function

4.14 Series Solutions Valid for Large Values of x

5. Boundary-Value Problems and Characteristic-Function Representations

5.1 Introduction

5.2 The Rotating String

5.3 The Rotating Shaft

5.4 Buckling of Long Columns Under Axial Loads

5.5 The Method of Stodola and Vianello

5.6 Orthogonality of Characteristic Functions

5.7 Expansion of Arbitrary Functions in Series of Orthogonal Functions

5.8 Boundary-Value Problems Involving Nonhomogeneous Differential Equations

5.9 Convergence of the Method of Stodola and Vianello

5.10 Fourier Sine Series and Cosine Series

5.11 Complete Fourier Series

5.12 Term-by-Term Differentiation of Fourier Series

5.13 Fourier-Bessel Series

5.14 Legendre Series

5.15 The Fourier Integral

6. Vector Analysis

6.1 Elementary Properties of Vectors

6.2 The Scalar Product of Two Vectors

6.3 The Vector Product of Two Vectors

6.4 Multiple Products

6.5 Differentiation of Vectors

6.6 Geometry of a Space Curve

6.8 The Vector Operator V

6.9 Differentiation Formulas

6.10 Line Integrals

6.11 The Potential Function

6.12 Surface Integrals

6.13 Interpretation of Divergence. The Divergence Theorem

6.14 Green's Theorem

6.15 Interpretation of Curl. Laplace's Equation

6.16 Stokes's Theorem

6.17 Orthogonal Curvilinear Coordinates

6.18 Special Coordinate Systems

6.19 Application to Two-Dimensional Incompressible Fluid Flow

6.20 Compressible Ideal Fluid Flow

7. Topics in Higher-Dimensional Calculus

7.1 Partial Differentiation. Chain Rules

7.2 Implicit Functions. Jacobian Determinants

7.3 Functional Dependence

7.4 Jacobians and Curvilinear Coordinates. Change of Variables in Integrals

7.5 Taylor Series

7.6 Maxima and Minima

7.7 Constraints and Lagrange Multipliers

7.8 Calculus of Variations

7.9 Differentiation of Integrals Involving a Parameter

7.10 Newton's Iterative Method

8. Partial Differential Equations

8.1 Definitions and Examples

8.2 The Quasi-Linear Equation of First Order

8.3 Special Devices. Initial Conditions

8.4 Linear and Quasi-Linear Equations of Second Order

8.5 Special Linear Equations of Second Order, with Constant Coefficients

8.6 Other Linear Equations

8.7 Characteristics of Linear First-Order Equations

8.8 Characteristics of Linear Second-Order Equations

8.9 Singular Curves on Integral Surfaces

8.10 Remarks on Linear Second-Order Initial-Value Problems

8.11 The Characteristics of a Particular Quasi-Linear Problem

9. Solutions of Partial Differential Equations of Mathematical Physics

9.1 Introduction

9.2 Heat Flow

9.3 Steady-State Temperature Distribution in a Rectangular Plate

9.4 Steady-State Temperature Distribution in a Circular Annulus

9.5 Poisson's Integral

9.6 Axisymmetrical Temperature Distribution in a Solid Sphere

9.7 Temperature Distribution in a Rectangular Parallelepiped

9.8 Ideal Fluid Flow about a Sphere

9.9 The Wave Equation. Vibration of a Circular Membrane

9.10 The Heat-Flow Equation. Heat Flow in a Rod

9.11 Duhamel's Superposition Integral

9.12 Traveling Waves

9.13 The Pulsating Cylinder

9.14 Examples of the Use of Fourier Integrals

9.15 Laplace Transform Methods

9.16 Application of the Laplace Transform to the Telegraph Equations for a Long Line

9.17 Nonhomogeneous Conditions. The Method of Variation of Parameters

9.18 Formulation of Problems

9.19 Supersonic Flow of ldeal Compressible Fluid Past an Obstacle

10. Functions of a Complex Variable

10.1 Introduction. The Complex Variable

10.2 Elementary Functions of a Complex Variable

10.3 Other Elementary Functions

10.4 Analytic Functions of a Complex Variable

10.5 Line Integrals of Complex Functions

10.6 Cauchy's Integral Formula

10.7 Taylor Series

10.8 Laurent Series

10.9 Singularities of Analytic Functions

10.10 Singularities at Infinity

10.11 Significance of Singularities

10.12 Residues

10.13 Evaluation of Real Definite Integrals

10.14 Theorems on Limiting Contours

10.15 Indented Contours

10.16 Integrals Involving Branch Points

11. Applications of Analytic Function Theory

11.1 Introduction

11.2 Inversion of Laplace Transforms

11.3 Inversion of Laplace Transforms with Branch Points. The Loop Integral

11.4 Conformal Mapping

11.5 Applications to Two-Dimensional Fluid Flow

11.6 Basic Flows

11.7 Other Applications of Conformal Mapping

11.8 The Schwarz-Christoffel Transformation

11.9 Green's Functions and the Dirichlet Problem

11.10 The Use of Conformal Mapping

11.11 Other Two-Dimensional Green's Functions