Mathematics and Basic Engineering Sciences
Mathematics and Basic Engineering Sciences underpins quantitative mechanical engineering; the exam tests calculus, linear algebra, ODEs, probability and statistics, numerical methods, and dimensional analysis as applied to engineering models.
Core rules and concepts
- A derivative is a rate; an integral accumulates. Use the chain rule for composite functions and check dimensions after every operation.
- Linear systems Ax=b are solvable when det A is nonzero; eigenvalues satisfy det(A-lambda I)=0 and govern stability of linear ODEs.
- A first-order linear ODE y'+P(x)y=Q(x) has integrating factor mu=e^(integral P dx); second-order systems are classified by damping ratio zeta.
- Numerical methods trade accuracy for effort: Newton-Raphson x_(n+1)=x_n-f/f' converges quadratically near a simple root but can diverge.
- Probability: means and variances add for independent variables; normal, binomial, and Poisson models have distinct assumptions.
Worked example
Solve y'+2y=4, y(0)=1. The integrating factor is mu=e^(integral 2 dx)=e^(2x). Then d/dx(y e^(2x))=4e^(2x); integrating gives y e^(2x)=2e^(2x)+C, so y=2+Ce^(-2x). Applying y(0)=1 gives 1=2+C, C=-1; hence y=2-e^(-2x). Check: y'=-2e^(-2x); y'+2y=-2e^(-2x)+4-2e^(-2x)=4. The solution satisfies both the ODE and the initial condition.
Traps
Using degrees in calculus, dropping the integration constant, treating a non-square matrix as invertible, and accepting a numerical answer without a convergence or residual check lose marks. Unit mismatches in empirical formulas are also common.