Mathematics I

Integrals

1. Real powers and calculus
2. ★★★☆☆ Intuition to some basic ideas of calculus
3. ★★☆☆☆ Understanding variable substitutions and domain splitting in integrals
4. ★★★☆☆ The correct multivariate mean-value theorem (no inequality)
5. ★★★☆☆ Trace, Laplacian, the Heat equation, divergence theorem
6. Green's theorem and differentiation under the integral sign

Fourier analysis

1. ★★☆☆☆ Fourier series and Hilbert spaces
2. ★★★☆☆ Discovering the Fourier transform
3. ★★★★☆ Limiting cases I: the integral of e^ax and the finite-domain Fourier transform
4. Take the derivative matrix on polynomials and make it continuous, i.e. extend it to an integral transform -- to demonstrate sF(s)

Differential equations

1. Introduction to differential equations (ways to think about: functional equations and recurrence relations, antiderivatives as basic example, algebraic equations)
2. existence and uniqueness non-crossingness of solutions, method of characteristics/hyperbolic PDEs/other DE stuff

Counter-examples series

1. ★★★★☆ Limiting cases I: the integral of e^ax and the finite-domain Fourier transform
2. ★★★★☆ Limiting cases II: repeated roots of a differential equation
3. ★★★★★ What's with e^(-1/x)? On smooth non-analytic functions: part I
4. What's with e^(-1/x)? On smooth non-analytic functions: part II

Geometry

1. A definition of geometry and trigonometry
2. Hyperbolic trigonometry (what about other conic sections?)

Inequalities or something

1. AM/GM and logarithms
2. Jensen's inequality and higher-order derivatives

Proof-writing and logical rigor

1. Introduction to proof-writing and rigour
2. Resolving some "paradoxes"
3. Reverse mathematics with elementary calculus
4. Real analysis and the importance of sup

Numerical mathematics

1. ★☆☆☆☆ Numerical linear algebra -- Matrix decompositions
2. Numerical arithmetic -- algorisms
3. Numerical analysis -- optimization etc.

Miscellaneous

1. ★★☆☆☆ Polynomial interpolation and the Vandermonde matrix
2. ★★★☆☆ Intuition to convergence
3. ★★★☆☆ Making sense of Euler's formula
4. Running, walking, yardsticks and Bezout's identity