Within the Natural Sciences Tripos at Cambridge, in first and second year there is a dedicated mathematics subject (which was my favourite subject in both years).

The Cambridge course is very much a Mathematical Methods course, focussing on creative application rather than formality.

Unlike for physics, I haven’t followed the structure of the official course handout in these notes. Instead, I have chosen to group together certain topics (for example, Sturm-Liouville theory with the other ODE stuff).

- Vector calculus
- Linear algebra
- Vector spaces
- Matrices
- Eigenvalues and eigenvectors
- Cartesia tensors

- Ordinary differential equations
- 2nd order ODEs
- Impulses and Green’s functions
- Series solutions to ODEs
- Sturm-Liouville Theory

- Partial differential equations
- Laplace’s and Poisson’s equations
- Method of images

- Calculus of Variations
- Fourier theory
- Fourier series
- Fourier transforms and convolutions
- Complex methods

- Analysis
- Power series and singularities
- Complex analysis and contour integration