Math
Mathematics for AI III · Chapter 8 — from vectors and norms to gradients, tangent planes, Hessians and extrema. Every topic has interactive lessons, a playground, practice in three difficulty levels and a mock short exam.
Vectors, norms & sequences
Scalar product and norms measure size and distance in ℝᵈ. With them, convergence of vector sequences reduces to convergence of each component. · Sheet 1
Functions & continuity
Functions of several variables: graphs, level sets, domains — and why continuity at a point has to be checked along every path, not just the axes. · Sheets 0 · 2
First derivatives
Partial derivatives, the gradient, the total derivative as best linear approximation, tangent planes and directional derivatives — and how they fit together. · Sheets 0 · 2 · 3
Second derivatives
The Hessian collects all second-order partial derivatives; Schwarz’s theorem says when the order of differentiation does not matter. · Sheets 0 · 3
Extrema
Stationary points, definiteness of the Hessian and the second-derivative test: minimum, maximum or saddle. · Sheet 4 · coming soon