Jacobian matrix

For $F : \R^n \to \R^m$, the $m \times n$ matrix of all partial derivatives $\partial F_i/\partial x_j$: the best linear approximation of $F$ near a point. Its determinant measures how $F$ stretches volumes.

When both input and output are vectors, "the derivative" has to say how each output responds to each input: an $m \times n$ table of rates. Arranged as a matrix it composes by multiplication (chain rule), inverts when $F$ is locally invertible, and its determinant gives the change-of-variables factor in integrals.

Zoom in on a smooth map until it looks linear: the Jacobian is that linear map. A tiny square at $x$ is mapped to a tiny parallelogram spanned by the columns of $J$; its area is $|\det J|$ times the square's. For a robot arm, column $j$ is how fast the hand moves when joint $j$ turns.

Inverse function theorem: if $m = n$ and $\det J_F(a) \ne 0$, $F$ is invertible near $a$ with $J_{F^{-1}} = J_F^{-1}$.

Polar coordinates $F(r, \theta) = (r\cos\theta, r\sin\theta)$: $J = \begin{pmatrix}\cos\theta & -r\sin\theta\\ \sin\theta & r\cos\theta\end{pmatrix}$, $\det J = r$. That $r$ is the familiar factor in $\dd x\,\dd y = r\,\dd r\,\dd\theta$.

Robotics lives on Jacobians (velocity kinematics, inverse kinematics, singularities where $\det J = 0$). Newton's method for systems inverts one per step. Normalizing flows need cheap $\log|\det J|$. Autodiff frameworks never build Jacobians explicitly: they compute Jacobian–vector (JVP) and vector–Jacobian (VJP) products.