**Calculus (mainly integration)**

- 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

+Often when you're first intrdouced to plotting functions, you wonder how it is that points on some plane defined by some law all happen to fit into this nice-looking curve. Calculus kinda gives you the intuition for it, allows you to accept it.

+Gradient -- path of steepest ascent: DE: z_x + z_y y' = |Del z| where S: z = z(x,y), C: z = f(x,y(x))

+Partial derivative intuition --momentum what's >p/>x and dp/dx if p(t) and t determines x? Draw some vector. To show intuition, draw 3D plot here and say that the full derivative is valid on a curve on the surface, and show the changes in the curve and stuff. field f(x(t),y(t),t) and think about the partial derivative of it wrt t

**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)

+generating functions and integral transforms

**Differential equations**

- Introduction to differential equations (ways to think about: functional equations and recurrence relations, antiderivatives as basic example, algebraic equations)

**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 and trigonometry**

- A beautiful way to think about geometry and trigonometry
- Hyperbolic trigonometry (what about other conic sections?)

**Inequalities or something**

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

**Miscellaneous**

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

## No comments:

## Post a Comment