High-school mathematics

Calculus (mainly integration) 
  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
+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
  1. Fourier series and Hilbert spaces
  2. Discovering the Fourier transform
  3. Limiting cases I: the integral of eax 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)
+generating functions and integral transforms

Differential equations
  1. 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
  1. Limiting cases I: the integral of eax 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 and trigonometry
  1. A beautiful way to think about 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
Miscellaneous

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