**Integrals**

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

**Fourier analysis**

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

**Differential equations**

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

**Counter-examples series**

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

**Geometry**

- A definition of geometry and trigonometry
- Hyperbolic trigonometry (what about other conic sections?)

**Inequalities or something**

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

**Proof-writing and logical rigor**

- Introduction to proof-writing and rigour
- Resolving some "paradoxes"
- Reverse mathematics with elementary calculus
- Real analysis and the importance of
*sup*

**Numerical mathematics**

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

**Miscellaneous**

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

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