Mathematics of theoretical physics

Physical theories and formulae are largely expressed through the language of mathematics, arguably the most effective quantitative language we have for the sciences. From the invention of calculus through to Einstein's Theory of General Relativity and the recent heavy use of mathematics in string theory, developments in mathematics and theoretical physics have been intimately intertwined since the time of the Renaissance.

Foundations

A strong mastery of basic high-school level algebra, trigonometry, analytic and synthetic geometry, and single-variable calculus is required at the very least if one wishes to do serious research in the physical sciences. Calculus is used extensively in Newtonian mechanics and gravity, for example with the second order linear differential equation F = ma.

Multivariable calculus

Multivariable Calculus is the extension of calculus to multiple variables. Gradients, curls, divergences, and all that are essential to understanding continuum theories and most notably Maxwell's equations for Electromagnetism.

Differential equations

Differential equations relate derivatives of functions and are used extensively for physical models. A few examples include:

Thermodynamics: the heat equation
Quantum Mechanics: the Schrodinger wave equation
Mechanics: Euler-Lagrange equations, action
- Noether's theorem: symmetry and conversation laws, Noether's current

Linear Algebra

Understanding crucial in quantum mechanics as well as in many classical mechanics applications involving rotation and moments of inertia, among other things.

Advanced

Group Theory

Used particularly in particle physics, parts of quantum mechanics, Noether's theorem

Differential Geometry

This forms the language of Einstein's theory of general relativity, which models spacetime as a four-dimensional spacetime Lorentzian manifold.

Mathematical tricks

Legendre Transformation: used in thermodynamics and classical mechanics to convert between Hamiltonian and Lagrangian formulations