Calculus in Motion, Interactively
The second volume - vector calculus, the Jacobian determinant, constrained optimization, matrix calculus and backprop, optimizers, probability, differential equations, manifolds and the calculus of variations, all carried by an arm that moves.
Volume I built the derivative and the integral of a function. Volume II lets the function move. A vector field assigns an arrow to every point; a line integral totals those arrows along a path; divergence and curl describe how the field spreads and spins; and one theorem - Green, Stokes and the divergence theorem, all the same statement - says the total on a boundary equals the total inside it. From there it is a short step to the Jacobian determinant, to constrained optimisation, and to the matrix calculus that makes backpropagation mechanical.
The last five parts are where this volume earns its keep elsewhere on the site: optimizers and the curvature that governs their convergence (handing off to Nonlinear Optimization), probability densities and the KL divergence, ODE integration, differentiation on a manifold of rotations, and optimising over whole trajectories. The first volume is the prerequisite; nothing here needs more than it taught.
The parts
A particle-flow canvas you edit by hand: arrows at every point, streamlines through them, and the two numbers that describe how a field changes.
Drag a path through a field and compare the work done; when the answer depends only on the endpoints, a potential surface is hiding underneath.
Drop a shrinking box probe and a paddlewheel into a field you shape, and read off outflow and rotation point by point.
Grow and shrink a region and watch the boundary total and the interior total stay equal - the Fundamental Theorem wearing three different costumes.
Deform a grid and watch cell areas scale by the determinant, the factor that makes a change of variables exact.
Double and triple integrals swept in polar, cylindrical and spherical coordinates, with the volume element each one brings.
Drag a constraint against the level sets of an objective and watch tangency appear as the condition for a constrained optimum, KKT inequalities included.
Layout conventions, a shape checker, and the handful of matrix-derivative identities that do most of the work in machine learning.
Build a small computation graph, run it forward, and watch the adjoints fill in backwards - the chain rule as an algorithm.
Race gradient descent, momentum and Adam on a surface you shape, and see how the condition number explains every zig-zag and slow crawl.
Push a Gaussian through a map, follow how densities transform with the Jacobian, and meet reparameterisation and the KL divergence.
A slope field you drop solutions into, Euler versus RK4 versus exact, and the pendulum and unicycle that turn into stability questions.
A naive update versus an exponential-map update on a sphere, and why rotations do not add - differentiation where the space is curved.
Drag two endpoints and watch the minimum-jerk path re-solve itself: optimising over whole functions, and the route to trajectory optimisation.