High School Mathematics

A typical American high school mathematics progression:

1. Algebra I — Linear equations, inequalities, functions, polynomials, and factoring.
2. Geometry — Euclidean geometry, proofs, transformations, area, volume, and the Pythagorean theorem.
3. Algebra II — Quadratic functions, complex numbers, logarithms, sequences, and conic sections.
4. Trigonometry — Trigonometric functions, identities, polar coordinates, and the unit circle.
5. Calculus (AP or introductory) — Limits, derivatives, integrals, and the fundamental theorem of calculus.

University Mathematics

Subjects that build on the high school foundation:

• Calculus II — Techniques of integration, sequences, series, and Taylor expansions.
• Calculus III (Multivariable Calculus) — Partial derivatives, multiple integrals, vector calculus, Green's, Stokes', and the Divergence theorems.
• Linear Algebra — Vectors, matrices, linear transformations, eigenvalues, and vector spaces.
• Differential Equations — ODEs, PDEs, Laplace transforms, and systems of equations.
• Discrete Mathematics — Logic, set theory, combinatorics, graph theory, and proof techniques — essential for computer science.
• Probability & Statistics — Random variables, distributions, hypothesis testing, and regression.
• Abstract Algebra — Groups, rings, fields, and the algebraic structures underlying symmetry.
• Real Analysis — Rigorous treatment of limits, continuity, differentiation, and integration on the real number line.
• Complex Analysis — Functions of a complex variable, contour integration, and residues.
• Number Theory — Primes, divisibility, modular arithmetic, and cryptography.