Engineering Mathematics and Fundamentals
Engineering Mathematics and Fundamentals underpins every later mechanical calculation: linear algebra for systems and transformations, calculus for rates and optimisation, differential equations for vibration and heat flow, Laplace transforms for transient response, and statistics for tolerances and test data. The exam favours setup and interpretation over long algebra.
Core rules
- A square system Ax = b has a unique solution when det A ≠ 0; eigenvalues follow det(A − λI) = 0.
- For y′ + p(t)y = q(t), the integrating factor is μ = e^(∫p dt); multiply both sides before integrating.
- Laplace derivatives: L{y′} = sY(s) − y(0) and L{y″} = s²Y(s) − sy(0) − y′(0).
- Distinguish population and sample standard deviation; the sample form uses n − 1.
- In numerical work, report units and check conditioning; small pivot values amplify round-off.
Worked example
Solve y′ + 2y = 4, y(0) = 1. Here p = 2, so μ = e^(2t). Then (y e^(2t))′ = 4e^(2t), giving y e^(2t) = 2e^(2t) + C, or y = 2 + Ce^(−2t). Applying y(0) = 1: 1 = 2 + C, so C = −1 and y = 2 − e^(−2t). Check: y′ = 2e^(−2t), so y′ + 2y = 2e^(−2t) + 4 − 2e^(−2t) = 4.
Traps
- Dropping the constant before applying the initial condition.
- Using degrees in trigonometric derivatives that require radians.
- Mixing population and sample formulas in variance.
- Treating a zero determinant as 'no solution' when it may be infinitely many.
- Rounding intermediate values too early in numerical iteration.