Mathematics for Undergraduate Physics
Maintained UY1 mathematical methods module surface: recommended core sequence, optional extensions, and notes-first assessment bridges into active UY1 module practice.
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 Phasors)Use 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 (Multivariable Calculus)Differentiate functions of several variables for gradients and Jacobians.
- Bernoulli Differential EquationsReduce a nonlinear Bernoulli equation to a linear one.
- Basics of Second-Order Differential EquationsSolve 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 Theorems)Use Gauss's and Stokes's theorems for field integrals.
- Matrices and Linear Algebra (Physics Basics)Use 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.
- Dimensional Analysis — 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.