Calculus for Machine Learning

2025-05-28 · Artintellica

Part I – Single‑Variable Foundations

#	Topic	Code‑Lab Highlights	Why ML cares
1	Limits & Continuity	zoom‐in plots, ε–δ checker	numerical stability, vanishing grads
2	Derivatives	finite diff vs autograd on `torch.sin`	gradients drive learning
3	Fundamental Theorem	trapezoid & Simpson vs `autograd.grad`	loss ↔ derivatives ↔ integrals
4	1‑D Optimization	hand‑rolled gradient descent	baby training loop
5	Taylor/Maclaurin	animated $e^{x}$ truncations	activation approx., positional encodings

Part II – Multivariable Core

#	Topic	Code‑Lab Highlights	Why ML cares
6	Vectors & ∇	quiver of ∇ $x^2+y^2$	visual back‑prop intuition
7	Jacobian & Hessian	tiny‑MLP Hessian spectrum	curvature, 2‑nd‑order opt.
8	Multiple Integrals	Monte‑Carlo 2‑D Gaussian	expected loss, ELBO
9	Change of Variables	affine flow, log‑det via autograd	flow‑based generative models
10	Line & Surface Integrals	streamplots, path work	RL trajectories, gradient flow

Part III – Vector Calculus & Differential Eqs.

#	Topic	Code‑Lab Highlights	Why ML cares
11	Divergence, Curl, Laplacian	heat‑equation on grid	diffusion models, graph Laplacian
12	ODEs	train Neural‑ODE on spirals	continuous‑time nets
13	PDEs	finite‑diff wave equation	physics‑informed nets, vision kernels

Part IV – Variations & Autodiff

#	Topic	Code‑Lab Highlights	Why ML cares
14	Functional Derivatives	gradient of $\int (f')^2\!dx$	weight decay as variational prob.
15	Back‑prop from Scratch	50‑line reverse‑mode engine	demystify autograd
16	Hessian‑Vector / Newton	SGD vs L‑BFGS, BFGS sketch	faster second‑order ideas

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