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 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)
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 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
  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

Miscellaneous

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