Mathematical Methods in Physics
Mathematical methods questions test vector resolution, trigonometry, differentiation and integration applied to motion, logarithms, and the interpretation of gradients and areas.
Core rules
- A vector of magnitude V at angle θ to the x-axis has components V cosθ and V sinθ. The dot product is ab cosθ; the cross-product magnitude is ab sinθ.
- Useful identities: sin²θ + cos²θ = 1; sin(A ± B) = sinA cosB ± cosA sinB.
- Differentiation: d/dt(tⁿ) = nt^(n−1); d/dt(sin ωt) = ω cos ωt; d/dt(cos ωt) = −ω sin ωt; d/dt(e^(kt)) = k e^(kt).
- Velocity is the gradient of an x–t graph; acceleration is the gradient of a v–t graph; displacement is the area under a v–t graph.
- Integration reverses differentiation; include the constant of integration for indefinite integrals.
- log(ab) = log a + log b; ln(e^x) = x; exponential decay becomes a straight line when ln y is plotted against t.
Worked example
For x(t) = 3t² − 2t + 1 in SI units, v = dx/dt = 6t − 2 and a = dv/dt = 6. At t = 2 s, v = 10 m/s and a = 6 m/s². The constant 1 affects position but not velocity or acceleration.
Traps
Forgetting the chain rule for sin(ωt) or e^(kt); dropping the constant of integration; using degrees in calculus formulas; treating a vector magnitude as a signed scalar; reading a gradient without its units.