Calculus for Machine Learning

2025-05-28 · Artintellica

Part I – Single‑Variable Foundations

#TopicCode‑Lab HighlightsWhy ML cares
1Limits & Continuityzoom‐in plots, ε–δ checkernumerical stability, vanishing grads
2Derivativesfinite diff vs autograd on torch.singradients drive learning
3Fundamental Theoremtrapezoid & Simpson vs autograd.gradloss ↔ derivatives ↔ integrals
41‑D Optimizationhand‑rolled gradient descentbaby training loop
5Taylor/Maclaurinanimated exe^{x} truncationsactivation approx., positional encodings

Part II – Multivariable Core

#TopicCode‑Lab HighlightsWhy ML cares
6Vectors & ∇quiver of ∇x2+y2x^2+y^2visual back‑prop intuition
7Jacobian & Hessiantiny‑MLP Hessian spectrumcurvature, 2‑nd‑order opt.
8Multiple IntegralsMonte‑Carlo 2‑D Gaussianexpected loss, ELBO
9Change of Variablesaffine flow, log‑det via autogradflow‑based generative models
10Line & Surface Integralsstreamplots, path workRL trajectories, gradient flow

Part III – Vector Calculus & Differential Eqs.

#TopicCode‑Lab HighlightsWhy ML cares
11Divergence, Curl, Laplacianheat‑equation on griddiffusion models, graph Laplacian
12ODEstrain Neural‑ODE on spiralscontinuous‑time nets
13PDEsfinite‑diff wave equationphysics‑informed nets, vision kernels

Part IV – Variations & Autodiff

#TopicCode‑Lab HighlightsWhy ML cares
14Functional Derivativesgradient of (f)2 ⁣dx\int (f')^2\!dxweight decay as variational prob.
15Back‑prop from Scratch50‑line reverse‑mode enginedemystify autograd
16Hessian‑Vector / NewtonSGD vs L‑BFGS, BFGS sketchfaster second‑order ideas

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