Mathematics and Numerical Methods
The FE mathematics section tests whether you can set up and solve engineering math quickly and recognize when a numerical method will misbehave. Expect algebra, trigonometry, analytic geometry, differential and integral calculus, linear algebra, differential equations, and numerical techniques such as root finding, interpolation, and integration.
Core rules
- Know derivatives and integrals of standard functions; the chain rule and integration by parts show up in centroid and work problems.
- Linear algebra: solve Ax = b by elimination; det(A) = 0 means no unique solution. Eigenvalues satisfy det(A − λI) = 0.
- First-order ODEs: separation of variables or integrating factor. Second-order linear ODEs use the characteristic equation, with under-, critically, and over-damped cases.
- Numerical integration: trapezoidal error is O(h²); Simpson's 1/3 rule requires an even number of intervals and is exact for cubics.
- Newton–Raphson: x(n+1) = x(n) − f(x(n))/f'(x(n)), with quadratic convergence near a simple root.
Worked example
Solve f(x) = x² − 2 starting at x₀ = 1, with f'(x) = 2x. x₁ = 1 − (1 − 2)/2 = 1.5. x₂ = 1.5 − (2.25 − 2)/3 = 1.41667. x₃ = 1.41667 − (2.00694 − 2)/2.83333 = 1.41422. The error roughly squares each iteration, so three steps give five correct digits.
Traps
- Calculator angle mode: a trig derivative or phasor result is wrong in the wrong mode.
- Simpson's 1/3 rule applied to an odd number of intervals.
- Newton–Raphson started where f' is near zero, or at an inflection point where it diverges.
- Mixing significant figures with decimal places; most FE answers are expected to three significant figures.