This book endeavours to strike a balance between mathematical and numerical coverage of a wide range of mathematical methods and numerical techniques. It strives to provide an introduction, especially for undergraduates and graduates, to engineering mathematics and its applications. Topics include advanced calculus, ordinary differential equations, partial differential equations, vector and tensor analysis, calculus of variations, integral equations, the finit difference method, reaction-diffusion system, and probability and statistics. The book also emphasizes the application of important mathematical methods with dozens of worked examples. The applied topics include elasticity, harmonic motion, chaos, kinematics, pattern formation and hypothesis testing. The book can serve as a textbook in engineering mathematics, mathematical modelling and scientific computing.

1 Calculus 1
1.1 Differentiations ....... 1
1.1.1 Definition . . . . . . 1
1.1.2 Differentiation Rules 2
1.1.3 In1plicit Differentiation . 4
1.2 Integrations . . . . . . . . . 5
1.2.1 Definition ........ 5
1.2.2 Integration by Parts .. 6
1.2.3 Taylor Series and Power Series 8
1.3 Partial Differentiation ........ 9
1.3.1 Partial Differentiation .... 9
1.3.2 Differentiation of an Integral 12
1.4 Multiple Integrals ..... 12
1.4.1 l1ultiple Integrals 12
1.4.2 Jacobian ...... 13
1.5 Some Special Integrals . . 16
1.5.1 Asymptotic Series 17
1.5.2 Gaussian Integrals 18
1.5.3 Error Functions . . 20
1.5.4 Gamma Functions 22
1.5.5 Bessel Functions 24
2 Vector Analysis 27
2.1 Vectors ... . . . . . . . . . . 27
2.1.1 Dot Product and Norm 28
v
CONTENTS
2.1.2 Cross Product
2.1.3 Vector Triple .
2.2 Vector Algebra . . . .
2.2.1 Differentiation of Vectors
2.2.2 Kinematics . . . . . . .
2.2.3 Line Integral . . . . . . .
2.2.4 Three Basic Operators ..
2.2.5 Son1e Important Theorems
2.3 Applications . . . . . . . . . .
2.3.1 Conservation of 1vlass
2.3.2 Saturn's Rings
3 Matrix Algebra
3.1 :rviatrix ....
3.2 Determinant.
3.3 Inverse . . . .
3.4 :rviatrix Exponential.
3.5 Hermitian and Quadratic Forms
3.6 Solution of linear systems
4 Complex Variables
4.1 Complex Numbers and Functions .
4.2 Hyperbolic Functions .
4.3 Analytic Functions
4.4 Complex Integrals . .
5 Ordinary Differential Equations
5.1 Introduction . . . . .
5.2 First Order ODEs ...
5.2.1 Linear ODEs ..
5.2.2 Nonlinear ODEs
5.3 Higher Order ODEs ..
5.3.1 General Solution
5.3.2 Differential Operator .
5.4 Linear System ...... .
5.5 Sturm-Liouville Equation
vi
5.5.1 Bessel Equation ........ .
5.5.2 Euler Buckling . . . . . . . . .
5.5.3 Nonlinear Second-Order ODEs
6 Recurrence Equations
6.1 Linear Difference Equations
6.2 Chaos and Dynamical Systems
6.2.1 Bifurcations and Chaos
6.2.2 Dynamic Reconstruction.
6.2.3 Lorenz Attractor ..
6.3 Self-similarity and Fractals ...
7 Vibration and Harmonic Motion
7.1 Undamped Forced Oscillations
7.2 Damped Forced Oscillations .
7.3 Normal Ivlodes . . . . . . . .
7.4 Small Amplitude Oscillations
8 Integral Transforms
8.1 Fourier Transform
8.1.1 Fourier Series . .
8.1.2 Fourier Integral .
8.1.3 Fourier Transform
8.2 Laplace Transforms.
8.3 ~avelet . . . . . . . . . .
9 Partial Differential Equations
9.1 First Order PDE
9.2 Classification
9.3 Classic PDEs . .
10 Techniques for Solving PDEs
10.1 Separation of Variables .
10.2 Transform l1ethods ... .
10.3 Similarity Solution ... .
10.4 Travelling ~ave Solution .
vii
10.5 Green's Function
10.6 Hybrid Method .
11 Integral Equations
11.1 Calculus of Variations ..... .
11.1.1 Curvature . . . . . . . . .
11.1.2 Euler-Lagrange Equation
11.1.3 Variations with Constraints
11.1.4 Variations for l1ultiple Variables
11.2 Integral Equations . . . . . . . .
11.2.1 Linear Integral Equations
11.3 Solution of Integral Equations .
11.3.1 Separable Kernels . .
11.3.2 Displacement Kernels
11.3.3 Volterra Equation
12 Tensor Analysis
12.1 Notations ..
12.2 Tensors . . .
12.3 Tensor Analysis .
13 Elasticity
13.1 Hooke's Law and Elasticity
13.2 l1axwell's Reciprocal Theorem
13.3 Equations of l1otion . . . . .
13.4 Airy Stress Functions ....
13.5 Euler-Bernoulli Beam Theory
14 Mathematical Models 201
14.1 Classic l1odels . . . . . . . . . . . . . . . . 201
14.1.1 Laplace's and Poisson's Equation . . 202
14.1.2 Parabolic Equation . . 202
14.1.3 Wave Equation . . . . . . 203
14.2 Other PDEs . . . . . . . . . . . . 203
14.2.1 Elastic Wave Equation . . 203
14.2.2 ltlaxwell's Equations . 204
viii
CONTENTS
14.2.3 Reaction-Diffusion Equation.
14.2.4 Fokker-Plank Equation
14.2.5 Black-Scholes Equation .
14.2.6 Schrodinger Equation ..
14.2.7 Navier-Stokes Equations .
14.2.8 Sine-Gordon Equation
15 Finite Difference Method 209
15.1 Integration of ODEs . . . 209
15.1.1 Euler Scheme . . . 210
15.1.2 Leap-Frog Jviethod . 212
15.1.3 Runge-Kutta Jviethod . 213
15.2 Hyperbolic Equations . . . . . 213
15.2.1 First-Order Hyperbolic Equation . 214
15.2.2 Second-Order Wave Equation . 215
15.3 Parabolic Equation . . 216
15.4 Elliptical Equation . . 218
16 Finite Volume Method 221
16.1 Introduction . . . . . . 221
16.2 Elliptic Equations . . 222
16.3 Parabolic Equations . 223
16.4 Hyperbolic Equations . 224
17 Finite Element Method 227
17.1 Concept of Elements . . . . . . . . 228
17.1.1 Simple Spring Systems . . . 228
17.1.2 Bar and Beam Elements . . 232
17.2 Finite Element Formulation . 235
17.2.1 Weak Formulation . 235
17.2.2 Galerkin Jviethod . 236
17.2.3 Shape Functions . . . 237
17.3 Elasticity . . . . . . . . . . 239
17.3.1 Plane Stress and Plane Strain . . 239
17.3.2 Implementation . . 242
17.4 Heat Conduction . . 244
ix
CONTENTS CONTENTS
17.4.1 Basic Formulation . . . . . . . . . . . 244
17 .4.2 Element-by-Element Assembly . . . . 246
17.4.3 Application of Boundary Conditions . 248
17.5 Time-Dependent Problems . . . 251
17.5.1 The Time Dimension. . . . . . 251
17.5.2 Time-Stepping . . . . . . . . . 253
17.5.3 1-D Transient Heat Transfer . . 253
17.5.4 Wave Equation .. . 254
18 Reaction Diffusion System 257
18.1 Heat Conduction Equation . 257
18.1.1 Fundamental Solutions . . 257
18.2 Nonlinear Equations . . . . 259
18.2.1 Travelling Wave . . . 259
18.2.2 Pattern Formation . . 260
18.3 Reaction-Diffusion System . . 263
19 Probability and Statistics 267
19.1 Probability . . . . . . . . . . . . . . . 267
19.1.1 Randomness and Probability . 267
19.1.2 Conditional Probability . . . . 275
19.1.3 Random Variables and Ivloments . 277
19.1.4 Binomial and Poisson Distributions. . 281
19.1.5 Gaussian Distribution . . . . . 283
19.1.6 Other Distributions . . . . . . 286
19.1.7 The Central Limit Theorem . . 287
19.2 Statistics . . . . . . . . . . . . . . . 289
19.2.1 Sample Ivlean and Variance . 290
19.2.2 Iviethod of Least Squares . 292
19.2.3 Hypothesis Testing . . 297
A Mathematical Formulas 311
A.1 Differentiations and Integrations . 311
A.2 Vectors and Matrices . . 312
A.3 Asymptotics . 314
A.4 Special Integrals . . . . 315
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