High-school mathematics

Review of elementary calculus, review of functions and elementary set theory, review of non-Cartesian co-ordinate systems, multivariable calculus, vector calculus, vector fields and their properties, Green's theorem, Gauss's theorem, Stokes' theorem, formal study of intuitively obvious theorems in calculus, differentiating under the integral sign, improper integrals, functional and differential equations, PDEs and their boundary conditions, stuff about transforming PDEs like hyperbolic PDEs, Jacobians, Green's functions and integral equations, method of characteristics, moments, generating functions and Fourier analysis and integral transforms, Cauchy formula for repeated integration; differential equations existence and uniqueness non-crossingness of solutions
  1. Intuition to some basic ideas of calculus
  2. Limiting cases I: the integral of eax and the finite-domain Fourier transform
  3. Limiting cases II: repeated roots of a differential equation
  4. Making sense of Euler's formula
  5. Intuition to convergence
  6. Discovering the Fourier transform
  7. Understanding variable substitutions and domain splitting in integrals
  8. Trace, Laplacian, the Heat equation, divergence theorem
  9. What's with e^(-1/x)? On smooth non-analytic functions: part I
  10. What's with e^(-1/x)? On smooth non-analytic functions: part II

  • The Calculus of Friendship (Steven Strogatz)

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