Essential Math Skills for Engineers

1 What Do Engineers Do?

2 Miscellaneous Math Skills.

2.1 Equations of Lines, Planes, and Circles.

2.2 Areas and Volumes of Common Shapes.

2.3 Roots of a Quadratic Equation.

2.4 Logarithms.

2.5 Reduction of Fractions and Lowest Common Denominators.

2.6 Long Division.

2.7 Trigonometry.

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

2.7.2 Areas of Triangles.

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

2.8 Complex Numbers and Algebra, and Euler’s Identity.

2.8.1 Solution of Differential Equations Having Sinusoidal Forcing Functions.

2.9 Common Derivatives and Their Interpretation.

2.10 Common Integrals and Their Interpretation.

2.11 Numerical Integration.

3 Solution of Simultaneous, Linear, Algebraic Equations.

3.1 How to Identify Simultaneous, Linear, Algebraic Equations.

3.2 The Meaning of a Solution.

3.3 Cramer’s Rule and Symbolic Equations.

3.4 Gauss Elimination.

3.5 Matrix Algebra.

4 Solution of Linear, Constant-Coeffi cient, Ordinary Differential Equations.

4.1 How to Identify Linear, Constant-Coeffi cient, Ordinary Differential Equations.

4.2 Where They Arise: The Meaning of a Solution.

4.3 Solution of First-Order Equations.

4.3.1 The Homogeneous Solution.

4.3.2 The Forced Solution for “Nice” f(t).

4.3.3 The Total Solution.

4.3.4 A Special Case.

4.4 Solution of Second-Order Equations.

4.4.1 The Homogeneous Solution.

4.4.2 The Forced Solution for “Nice” f(t).

4.4.3 The Total Solution.

4.4.4 A Special Case.

4.5 Stability of the Solution.

4.6 Solution of Simultaneous Sets of Ordinary Differential Equations with the Differential Operator.

4.6.1 Using the Differential Operator to Verify Solutions.

4.7 Numerical (Computer) Solutions.

5 Solution of Linear, Constant-Coeffi cient, Difference Equations.

5.1 Where Difference Equations Arise.

5.2 How to Identify Linear, Constant-Coeffi cient Difference Equations.

5.3 Solution of First-Order Equations.

5.3.1 The Homogeneous Solution.

5.3.2 The Forced Solution for “Nice” f(n).

5.3.3 The Total Solution.

5.3.4 A Special Case.

5.4 Solution of Second-Order Equations.

5.4.1 The Homogeneous Solution.

5.4.2 The Forced Solution for “Nice” f(n).

5.4.3 The Total Solution.

5.4.4 A Special Case.

5.5 Stability of the Solution.

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

5.6.1 Using the Difference Operator to Verify Solutions.

6 Solution of Linear, Constant-Coeffi cient, Partial Differential Equations.

6.1 Common Engineering Partial Differential Equations.

6.2 The Linear, Constant-Coeffi cient, Partial Differential Equation.

6.3 The Method of Separation of Variables.

6.4 Boundary Conditions and Initial Conditions.

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

7 The Fourier Series and Fourier Transform.

7.1 Periodic Functions.

7.2 The Fourier Series.

7.3 The Fourier Transform.

8 The Laplace Transform.

8.1 Transforms of Important Functions.

8.2 Useful Transform Properties.

8.3 Transforming Differential Equations.

8.4 Obtaining the Inverse Transform Using Partial Fraction Expansions.

9 Mathematics of Vectors.

9.1 Vectors and Coordinate Systems.

9.2 The Line Integral.

9.3 The Surface Integral.

9.4 Divergence.

9.4.1 The Divergence Theorem.

9.5 Curl.

9.5.1 Stokes’ Theorem.

9.6 The Gradient of a Scalar Field.

Index.