Mathematical methods for first-year university physics
Build the mathematical tools for first-year university physics with an ordered core sequence, optional extensions and links to relevant practice.
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Before you begin
These lessons build the mathematical methods for first-year physics. The aim is to pick the right tool quickly: gradients for field problems, differential equations for dynamics and transients, and complex numbers for phase and A.C. systems.
Learning goals
- Use complex numbers, linear algebra, and differential equations to solve coupled physical models.
- Use vector and differential or integral calculus to express physical change and accumulation.
- Choose an efficient mathematical method and validate the result physically.
Lessons
Work through them in order.
- Complex numbers, Euler’s formula and phasorsUse complex numbers for phasors and amplitude–phase algebra.
- Differentiation techniquesDifferentiate fluently, including the chain rule.
- Integration techniquesChoose and apply the main integration methods.
- Integration without integrationUse a shortcut for repeated trigonometric–exponential integrals.
- Partial derivatives and multivariable calculusDifferentiate functions of several variables for gradients and Jacobians.
- Bernoulli differential equationsReduce a nonlinear Bernoulli equation to a linear one.
- Second-order differential equations: the basicsSolve constant-coefficient second-order equations from the characteristic equation, with and without forcing.
- First-order differential equationsSolve first-order equations for RC, drag and relaxation models.
- First-order linear differential equationsSolve first-order linear equations with an integrating factor.Supporting
- Second-order differential equationsSolve second-order equations for oscillating systems.
- Steps for solving first-order differential equationsFollow a standard procedure for solving first-order equations.Supporting
- Separable differential equationsSolve separable equations by direct integration.
- Vector calculus integral theoremsUse Gauss's and Stokes's theorems for field integrals.
- Matrices and linear algebra for physicsUse matrices and eigenvalues for coupled systems.
Practise and check
Topic reference
Quick references
- Table of derivatives — Derivative recall.
- Integration table — Antiderivative recall.
- Trigonometry — Identity refresh and phase manipulations.
- Taylor series — Approximation checks.
- Hyperbolic functions — Differential-equation solution forms.
- SI units, prefixes and dimensional checks — Equation sanity checks.
- Uncertainty & error propagation — Measurement/result reporting.
- Euler-Lagrange equation — A Year 2 topic beyond UY1.
Method-selection checkpoint
Choose a method before opening the answer. For each prompt, state the mathematical tool, the physical reason it fits, and one check you would make on the result.
- A cooling model obeys dT/dt = -k(T-Tᵣₒₒₘ). You need T(t) from an initial temperature.
- A spherically symmetric electric field is known on a closed surface. You need the enclosed charge.
- Two coupled small oscillations are written as Mx double dot + Kx = 0. You need the normal-mode frequencies.
Checkpoint answers — compare the decision, not just the algebra
- Use separation of variables or a first-order linear ODE method. The temperature difference is the dependent quantity; check the initial condition, units of k, and the long-time limit T → Tᵣₒₒₘ.
- Use Gauss’s law as a surface integral. Symmetry makes the flux integral efficient; check the surface orientation, dimensions, and whether the assumed symmetry really makes E constant on the surface.
- Use linear algebra and an eigenvalue condition such as det(K-ω²M) = 0. Check that ω² has units of inverse time squared and that identical uncoupled oscillators recover the expected limiting frequencies.