Essential Math Skills for Engineers Book

Preface xi

1 What Do Engineers Do? 1

2 Miscellaneous Math Skills 15

2.1 Equations of Lines, Planes, and Circles 15

2.2 Areas and Volumes of Common Shapes 21

2.3 Roots of a Quadratic Equation 22

2.4 Logarithms 23

2.5 Reduction of Fractions and Lowest Common Denominators 24

2.6 Long Division 26

2.7 Trigonometry 27

2.7.1 The Common Trigonometric Functions: Sine, Cosine, and Tangent 27

2.7.2 Areas of Triangles 34

2.7.3 The Hyperbolic Trigonometric Functions: Sinh, Cosh, and Tanh 36

2.8 Complex Numbers and Algebra, and Euler's Identity 39

2.8.1 Solution of Differential Equations Having Sinusoidal Forcing Functions 46

2.9 Common Derivatives and Their Interpretation 48

2.10 Common Integrals and Their Interpretation 51

2.11 Numerical Integration 56

3 Solution of Simultaneous, Linear, Algebraic Equations 59

3.1 How to Identify Simultaneous, Linear, Algebraic Equations 59

3.2 The Meaning of a Solution 60

3.3 Cramer's Rule and Symbolic Equations 61

3.4 Gauss Elimination 67

3.5 Matrix Algebra 68

4 Solution of Linear, Constant-Coefficient, Ordinary Differential Equations 83

4.1 How to Identify Linear, Constant-Coefficient, Ordinary Differential Equations 83

4.2 Where They Arise: The Meaning of a Solution 85

4.3 Solution of First-Order Equations 86

4.3.1 The Homogeneous Solution 87

4.3.2 The Forced Solution for "Nice" f(t) 88

4.3.3 The Total Solution 90

4.3.4 A Special Case 94

4.4 Solution of Second-Order Equations 96

4.4.1 The Homogeneous Solution 96

4.4.2 The Forced Solution for "Nice" f(t) 99

4.4.3 The Total Solution 99

4.4.4 A Special Case 102

4.5 Stability of the Solution 105

4.6 Solution of Simultaneous Setsof Ordinary Differential Equations with the Differential Operator 105

4.6.1 Using the Differential Operator to Verify Solutions 110

4.7 Numerical (Computer) Solutions 113

5 Solution of Linear, Constant-Coefficient, Difference Equations 119

5.1 Where Difference Equations Arise 119

5.2 How to Identify Linear, Constant-Coefficient Difference Equations 122

5.3 Solution of First-Order Equations 124

5.3.1 The Homogeneous Solution 125

5.3.2 The Forced Solution for "Nice" f(n) 125

5.3.3 The Total Solution 127

5.3.4 A Special Case 130

5.4 Solution of Second-Order Equations 131

5.4.1 The Homogeneous Solution 131

5.4.2 The Forced Solution for "Nice" f(n) 134

5.4.3 The Total Solution 134

5.4.4 A Special Case 138

5.5 Stability of the Solution 140

5.6 Solution of Simultaneous Sets of Difference Equations with the Difference Operator 141

5.6.1 Using the Difference Operator to Verify Solutions 144

6 Solution of Linear, Constant-Coefficient, Partial Differential Equations 149

6.1 Common Engineering Partial Differential Equations 149

6.2 The Linear, Constant-Coefficient, Partial Differential Equation 150

6.3 The Method of Separation of Variables 151

6.4 Boundary Conditions and Initial Conditions 153

6.5 Numerical (Computer) Solutions via Finite Differences: Conversion to Difference Equations 161

7 The Fourier Series and Fourier Transform 169

7.1 Periodic Functions 171

7.2 The Fourier Series 174

7.3 The Fourier Transform 181

8 The Laplace Transform 187

8.1 Transforms of Important Functions 188

8.2 Useful Transform Properties 191

8.3 Transforming Differential Equations 194

8.4 Obtaining the Inverse Transform Using Partial Fraction Expansions 196

9 Mathematics of Vectors 207

9.1 Vectors and Coordinate Systems 207

9.2 The Line Integral 211

9.3 The Surface Integral 214

9.4 Divergence 216

9.4.1 The Divergence Theorem 218

9.5 Curl 220

9.5.1 Stokes' Theorem 223

9.6 The Gradient of a Scalar Field 225

Index 229