Mathematics and Physics from the Ground Up
PLEN

Mathematics

From a pre-course to partial differential equations. Every lesson starts with a question from physics or engineering and ends with exercises and answers.

Part 0 Pre-course

  1. 0.1Numbers, powers, logarithms and elementary functions ready
  2. 0.2Equations, inequalities, polynomials and partial fractions ready requires: 0.1
  3. 0.3Trigonometry and hyperbolic functions ready requires: 0.1
  4. 0.4Analytic geometry and conic sections ready requires: 0.2, 0.3
  5. 0.5Units, dimensional analysis, estimation and measurement uncertainty ready requires: 0.1, 0.2, 0.3

Part I Vectors and single-variable calculus

  1. 1.1Vectors ready requires: 0.3, 0.4
  2. 1.2Sequences, limits and continuity ready requires: 0.1, 0.2, 0.3
  3. 1.3Derivatives ready requires: 1.1, 1.2
  4. 1.4Differential calculus in practice ready requires: 0.5, 1.3
  5. 1.5Integrals ready requires: 1.4
  6. 1.6Series and power series ready requires: 1.2, 1.5
  7. 1.7Complex numbers and phasors ready requires: 0.2, 0.3, 0.4, 1.6

Part II Linear algebra

  1. 2.1Matrices, determinants and linear systems ready requires: 0.2, 1.1
  2. 2.2Linear maps: rotations, change of basis, Lorentz transformation ready requires: 0.3, 1.7, 2.1
  3. 2.3Eigenvalues and eigenvectors ready requires: 1.7, 2.2

Part III Multivariable calculus

  1. 3.1Multivariable chain rule, extrema and Lagrange multipliers draft requires: 1.1, 1.4, 2.3
  2. 3.2Polar, cylindrical and spherical coordinates draft requires: 0.3, 3.1
  3. 3.3Double and triple integrals draft requires: 1.5, 2.1, 3.2
  4. 3.4Line integrals, conservative fields and exact differentials draft requires: 3.1, 3.3
  5. 3.5Flux, divergence and curl ready requires: 3.4
  6. 3.6The theorems of Green, Stokes and Gauss ready requires: 3.5

Part IV Differential equations and integral transforms

  1. 4.1Ordinary differential equations draft requires: 1.5, 1.7
  2. 4.2Fourier series draft requires: 1.6, 4.1
  3. 4.3The Fourier transform and the Dirac delta draft requires: 4.2
  4. 4.4The Laplace transform draft requires: 0.2, 4.3
  5. 4.5Linear systems and state space draft requires: 2.3, 4.4
  6. 4.6Nonlinear systems, phase plane and stability draft requires: 3.1, 4.5
  7. 4.7PDEs: the wave, diffusion and Laplace equations ready requires: 3.2, 3.5, 4.2

Part V Probability and data

  1. 5.1Probability and distributions draft requires: 1.6, 3.3
  2. 5.2Measurement statistics and curve fitting draft requires: 0.5, 2.1, 3.1, 5.1

Part VI Toolkit

  1. 6.1Calculus of variations ready requires: 1.5, 3.1, 4.1
  2. 6.2Numerical methods: Python and C++ ready requires: 1.4, 4.1