# Book:A.J.M. Spencer/Engineering Mathematics/Volume I

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## A.J.M. Spencer:

## A.J.M. Spencer: *Engineering Mathematics, Volume $\text { I }$*

Published $\text {1977}$, **Van Nostrand Reinhold**

### Summary

Often referred to as **Spencer et al**, this work was written as a collaboration of nine: A.J.M. Spencer, D.F. Parker, D.S. Berry, A.H. England, T.R. Faulkner, W.A. Green, J.T. Holden, D. Middleton and T.G. Rogers.

From the preface:

- "Each chapter was first drafted by one or two authors, read and checked by several others, and then discussed and amended until a mutually agreed version was produced. The text is therefore a truly collaborative effort; no part of it is the sole work of any individual, and we share responsibility for the whole."

Be that as it may, Spencer's name is the first on the list.

The author of this page used this book to good effect in his undergraduate days.

### Contents

- Preface
- The Greek Alphabet
- CHAPTER 1. ORDINARY DIFFERENTIAL EQUATIONS
- 1.1 Introduction
- 1.2 Geometrical Interpretation of Solutions of Ordinary Differential Equations
- 1.3 First-order Equations
- 1.4 Linear Ordinary Differential Equations with Constant Coefficients. D Operator Notation
- 1.5 Solution of Homogeneous Linear Equations with Constant Coefficients
- 1.6 Theory of Damped Free Vibrations
- 1.7 Inhomogeneous Second-order Equations with Constant Coefficients
- 1.8 Theory of Forced Vibrations
- 1.9 Simultaneous Linear Differential Equations with Constant Coefficients
- 1.10 Euler's Equation
- Problems
- Bibliography

- CHAPTER 2. FOURIER SERIES
- 2.1 Introduction
- 2.2 Derivation of the Fourier Series
- 2.3 Convergence of Fourier Series
- 2.4 Fourier Sine and Cosine Series
- 2.5 Integration and Differentiation of Fourier Series
- 2.6 Application of Fourier Series
- Problems

- CHAPTER 3. LAPLACE TRANSFORMS
- 3.1 Introduction
- 3.2 Transforms of Derivatives
- 3.3 Step Function and Delta Function
- 3.4 Properties of the Laplace Transform
- 3.5 Linear Ordinary Differential Equations
- 3.6 Difference and Integral Equations
- 3.7 Some Physical Problems
- Problems
- Bibliography

- CHAPTER 4. PARTIAL DIFFERENTIATION, WITH APPLICATIONS
- 4.1 Basic Results
- 4.2 The Chain Rule and Taylor's Theorem
- 4.3 Total Derivatives
- 4.4 Stationary Points
- 4.5 Further Applications
- Problems
- Bibliography

- CHAPTER 5. MULTIPLE INTEGRALS
- 5.1 Multiple Integrals and Ordinary Integrals
- 5.2 Evaluation of Double Integrals
- 5.3 Triple Integrals
- 5.4 Line Integrals
- 5.5 Surface Integrals
- Problems
- Bibliography

- CHAPTER 6. VECTOR ANALYSIS
- 6.1 Introduction
- 6.2 Vector Functions of One Variable
- 6.3 Scalar and Vector Fields
- 6.4 The Divergence Theorem
- 6.5 Stokes's Theorem
- 6.6 The Formulation of Partial Differential Equations
- 6.7 Orthogonal Curvilinear Coordinates
- Problems
- Bibliography

- CHAPTER 7. PARTIAL DIFFERENTIAL EQUATIONS
- 7.1 Introduction
- 7.2 The One-dimensional Wave Equation
- 7.3 The Method of Separation of Variables
- 7.4 The Wave Equation
- 7.5 The Heat Conduction and Diffusion Equation
- 7.6 Laplace's Equation
- 7.7 Laplace's Equation in Cylindrical and Spherical Polar Coordinates
- 7.8 Inhomogeneous Equations
- 7.9 General Second-order Equations
- Problems
- Bibliography

- CHAPTER 8. LINEAR ALGEBRA - THEORY
- 8.1 Systems of Linear Algebraic Equations. Matrix Notation
- 8.2 Elementary Operations of Matrix Algebra
- 8.3 Determinants
- 8.4 The Inverse of a Matrix
- 8.5 Orthogonal Matrices
- 8.6 Partitioned Matrices
- 8.7 Inhomogeneous Systems of Linear Equations
- 8.8 Homogeneous Systems of Liacar Equations
- 8.9 Eigenvalues and Eigenvectors
- Problems
- Bibliography

- CHAPTER 9. INTRODUCTION TO NUMERICAL ANALYSIS
- 9.1 Numerical Approximation
- 9.2 Evaluation of Formulae
- 9.3 Flow Diagrams or Charts
- 9.4 Solution of Single Algebraic and Transcendental Equations
- Problems
- Bibliography

- CHAPTER 10. LINEAR ALGEBRA - NUMERICAL METHODS
- 10.1 Introduction
- 10.2 Direct Methods for the Solution of Linear Equations
- 10.3 Iterative Methods for the Solution of Linear Equations
- 10.4 Numerical Methods of Matrix Inversion
- 10.5 Eigenvalues and Eigenvectors
- Problems
- Bibliography

- CHAPTER 11 FINITE DIFFERENCES
- 11.1 Introduction
- 11.2 Finite Differences and Difference Tables
- 11.3 Interpolation
- 11.4 Numerical Integration
- 11.5 Numerical Differentiation
- Problems
- Bibliography

- CHAPTER 12. ELEMENTARY STATISTICS - PROBABILITY THEORY
- 12.1 Introduction
- 12.2 Probability and Equi-likely Events
- 12.3 Probability and Relative Frequency
- 12.4 Probability and Set Theory
- 12.5 The Random Variable
- 12.6 Basic Variates
- 12.7 Bivariate and Multivariate Probability Distributions
- 12.8 Simulation and Monte Carlo Methods
- Problems
- Bibliography

- Appendix
- Table A1 : Laplace Transforms
- Table A2 : The StandardiZed Normal Variate

- Answers to Exercises and Problems

- Index

**CONTENTS OF VOLUME 2**- Chapter 1. Linear Programming
- Chapter 2. Non-linear and Dynamic Programming
- Chapter 3. Further Statistics - Estimation and Inference
- Chapter 4. Complex Variables
- Chapter 5. Integral Transforms
- Chapter 6. Ordinary Differential Equations
- Chapter 7. Numerical Solution of Differential Equations
- Chapter 8. Variational Methods

## Source work progress

- 1977: A.J.M. Spencer:
*Engineering Mathematics: Volume $\text { I }$*... (previous) ... (next): Chapter $1$ Ordinary Differential Equations: $1.1$ Introduction: Classification of Differential Equations