Mathematics
What to master
The FE mathematics scope is broad but shallow: vector algebra (dot and cross products, projections, unit vectors), analytic geometry (lines, planes, conics, distance formulas), single- and multivariable calculus (partial derivatives, multiple integrals, divergence, curl, line and surface integrals), linear algebra (matrix arithmetic, determinants, inverse, eigenvalues, solving linear systems by elimination), ordinary differential equations (separable, first-order linear with integrating factor, second-order constant-coefficient), and numerical methods (Newton-Raphson, trapezoidal and Simpson's rules, Euler's method).
Traps
- Mixing degrees and radians: derivatives and series for trig functions assume radians.
- Confusing gradient (scalar in, vector out), divergence (vector in, scalar out) and curl (vector in, vector out).
- Sign errors when reversing definite-integral limits, and forgetting the constant of integration.
- Assuming a matrix is invertible: check that det(A) is nonzero; eigenvalues come from det(A - lambda*I) = 0.
- Newton-Raphson diverging or oscillating near a zero derivative -- the exam expects you to recognize when the method is unreliable.
Worked example
For the vector field F = (x^2, 3y, -z), the divergence is the sum of partials: d/dx(x^2) + d/dy(3y) + d/dz(-z) = 2x + 3 - 1 = 2x + 2. If a choice offers 2x + 3 + 1, you have been caught by the sign on the z-term.
Rule of thumb: check every antiderivative by differentiating it back, and sanity-check the magnitude and units of any numerical answer before choosing an option.