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Advanced Calculus Topics

Three tracks for after Calculus I-III: differential equations, calculus for machine learning (gradient descent, backprop, autodiff), and a rigorous bridge to real analysis.

advanced Free calculusmathdifferential-equationsmachine-learning
📖 25 readings ⚡ 25 exercises · 259 min total
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3 modules · 25 lessons
13. Differential Equations 10 lessons · 95 min
  1. 1 What a differential equation is; slope fields and Euler's method 10 min
  2. 2 Separable equations 9 min
  3. 3 First-order linear equations and the integrating factor 9 min
  4. 4 Exact equations 9 min
  5. 5 Modelling: growth, decay, logistic, and Newton's cooling 9 min
  6. 6 Second-order linear equations with constant coefficients 9 min
  7. 7 Nonhomogeneous equations: undetermined coefficients and variation of parameters 10 min
  8. 8 Series solutions 9 min
  9. 9 Systems of linear equations and the phase plane 11 min
  10. 10 Adv: Laplace transforms 10 min
14. Calculus for Machine Learning 7 lessons · 76 min
  1. 11 Gradient descent: calculus turned into an algorithm 11 min
  2. 12 Convexity, and why it makes optimization tractable 11 min
  3. 13 Jacobians and Hessians: derivatives as matrices 10 min
  4. 14 The chain rule as backpropagation 11 min
  5. 15 Automatic differentiation: forward and reverse mode 11 min
  6. 16 Matrix calculus for least squares and logistic regression 11 min
  7. 17 Constrained optimization: from Lagrange to KKT 11 min
15. Adv: Bridge to Real Analysis 8 lessons · 88 min
  1. 18 Adv: the real numbers — completeness and the least upper bound 10 min
  2. 19 Adv: Cauchy sequences and Bolzano–Weierstrass 11 min
  3. 20 Adv: rigorous ε–δ — proving the limit laws and the derivative rules 10 min
  4. 21 Adv: uniform continuity 11 min
  5. 22 Adv: the Riemann integral — definition and criteria for integrability 12 min
  6. 23 Adv: sequences of functions — pointwise vs. uniform convergence 10 min
  7. 24 Adv: metric spaces — open, closed, compact, complete 12 min
  8. 25 Adv: the derivative as a linear map; the inverse and implicit function theorems 12 min

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