The Jacobian Matrix

When we change coordinates on a manifold, vectors and tensors transform. The transformation is governed by the Jacobian — the matrix of all partial derivatives of the coordinate change.

For a map $F: \mathbb{R}^n \to \mathbb{R}^m$ with $F = (F_0, F_1, \ldots, F_{m-1})$, the Jacobian at a point $p$ is:

$J_{ij} = \frac{\partial F_i}{\partial x_j}(p)$

Example: Polar-to-Rectangular

The map $F(r, \theta) = (r\cos\theta,\; r\sin\theta)$ has Jacobian:

$J = \begin{pmatrix} \cos\theta & -r\sin\theta \\ \sin\theta & r\cos\theta \end{pmatrix}$

At $(r=1, \theta=0)$: $J = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ — the identity.

At $(r=2, \theta=0)$: $J = \begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix}$ — $\partial y / \partial \theta = r = 2$.

Determinant

The determinant $\det(J)$ is the local volume scale factor: how much the map stretches areas. For polar coordinates, $\det(J) = r$, so the area element in polar coords is $r\,dr\,d\theta$.

Your Task

Implement jacobian(F, n) that takes a vector-valued function $F: \mathbb{R}^n \to \mathbb{R}^m$ and returns a function J(point) that computes the Jacobian matrix $J[i][j] = \partial F_i / \partial x_j$ at that point.