Engineering Mathematics
Engineering Mathematics is tested as a toolkit for metallurgical modelling: matrix methods for mass balances, calculus for transport gradients, ODEs for kinetics, probability for data, and numerical methods for equations without closed forms. Questions often embed the mathematics in a process context, so set up the equation before computing.
Core rules
- A linear system is consistent only when rank(A) equals rank([A|b]); equal ranks with fewer equations than unknowns mean infinite solutions.
- For a level surface f(x,y,z)=c, grad f is normal; divergence measures flux per volume and curl measures circulation.
- Constant-coefficient linear ODEs are solved by characteristic roots; repeated roots need a t e^{rt} term. Laplace transforms handle initial conditions algebraically.
- Trapezoidal integration has error O(h^2); Simpson's rule needs an even number of intervals and is O(h^4) for smooth integrands.
- Conditional probability uses P(A|B)=P(A∩B)/P(B); do not reverse the conditioning.
Worked example
Check consistency of x+y=2 and 2x+2y=5. The coefficient matrix A=[[1,1],[2,2]] has rank 1. The augmented matrix [[1,1,2],[2,2,5]] has a second row that is not a multiple of the first, so its rank is 2. Since ranks differ, the equations contradict: y=2−x gives 2x+2(2−x)=4, not 5. No solution exists.
Traps
- Using Simpson's rule with an odd number of subintervals.
- Forgetting the Jacobian when changing variables in multiple integrals.
- Reporting an eigenvector without checking A v = λ v.
- Treating a small numerical residual as proof of consistency; rank is exact.