Mathematical Methods and Laboratory/Experimental Physics
Core concepts
This domain blends mathematical technique with experimental practice. Know vector calculus (gradient, divergence, curl), linear algebra (eigenvalues, matrices, orthogonality), differential equations (first-order separable, second-order linear with constant coefficients), Fourier series and transforms, and complex numbers. On the laboratory side, understand uncertainty propagation, significant figures, calibration, systematic versus random error, and the design of controlled experiments.
Traps
- Uncertainty propagation: for independent random errors, quadrature add: σ_f = √(Σ(∂f/∂xᵢ σᵢ)²); do not simply add absolute errors.
- Systematic errors do not shrink with more data; only random errors average down.
- Fourier conventions differ by factors of 2π; check the definition used.
- Eigenvalues of a real symmetric matrix are real and eigenvectors orthogonal - a useful check.
- Dimensional analysis catches algebra slips in differential equations.
Worked example
A quantity f = xy is measured with x = 4.0 ± 0.1 and y = 3.0 ± 0.1. Relative errors are 2.5% and 3.3%; combined in quadrature, √(0.025² + 0.033²) ≈ 4.1%, so f = 12.0 ± 0.5. Adding absolute errors (0.2) would overstate the uncertainty; the correct treatment combines relative errors for a product.
Rule of thumb
Always report an uncertainty with units and state whether errors are random or systematic; a result without an uncertainty is incomplete.