Calculus & Analysis

Read these first

1. 01Taylor Series Expansion

   Any smooth function looks like a straight line up close, a parabola if you zoom out slightly less, and a cubic if you keep going, the Taylor series is the recipe for building that approximation, one derivative at a time.

2. 02The Fourier Transform

   Any wiggly signal, however messy, is a sum of plain sine waves at different frequencies, the Fourier transform is the recipe for reading off exactly which frequencies are in it and how strong each one is.

3. 03The Heat Equation and Diffusion

   The equation that describes heat spreading through a metal bar is, after a change of variables, exactly the Black-Scholes equation, both describe a quantity smoothing itself out over time at a rate set by curvature.

Then the rest

1. 04Derivatives and Differentiation Rules
2. 05Gradients and Directional Derivatives
3. 06Improper Integrals
4. 07Limits and Continuity
5. 08Ordinary Differential Equations
6. 09Integration Techniques
7. 10Metric Spaces and Completeness
8. 11PDE Classification and Boundary Conditions
9. 12Sequences and Series Convergence
10. 13The Fundamental Theorem of Calculus
11. 14The Hessian and Second-Order Behavior
12. 15The Jacobian and Change of Variables
13. 16The Laplace Transform
14. 17The Mean Value Theorem
15. 18Asymptotic Expansions and the Saddle-Point Method
16. 19Calculus of Variations and the Euler-Lagrange Equation
17. 20Complex Analysis and Analytic Functions
18. 21Hilbert Spaces and L2
19. 22The Residue Theorem and Contour Integration