Jacobian Matrix

Pack all the partials into one matrix; its determinant is the local stretch.

The idea

The Jacobian matrix of a map $f \colon \mathbb{R}^{n} \to \mathbb{R}^{m}$ collects all of its first partial derivatives into the single object that plays the role the derivative number plays in one variable.

Definition (Jacobian matrix).

The Jacobian matrix of $f \colon \mathbb{R}^{n} \to \mathbb{R}^{m}$ is the $m \times n$ matrix $J = \left[ \frac{\partial f_i}{\partial x_j} \right],$ whose $(i,j)$ entry is the partial derivative of the $i$-th output with respect to the $j$-th input.

Row $i$ holds every partial of the single output $f_i$; column $j$ records the effect of moving the single input $x_j$. This is the total derivative written in coordinates: when $f$ is differentiable at $a$, the linear map approximating $f$ near $a$ is multiplication by $J$ evaluated at $a$.

Differentiability here is a genuine hypothesis, because possessing all the partials is strictly weaker than being differentiable: partials probe $f$ only along the coordinate directions, and a map can have every one of them at a point without being differentiable there. The following theorem closes the gap.

Theorem (The $C^{1}$ theorem).

If all the partial derivatives of $f$ exist on an open set containing $a$ and are continuous at $a$, then $f$ is differentiable at $a$, with $J(a)$ as its derivative.

When $n = m$ the matrix is square, and $\det J$ measures how much $f$ stretches area or volume near the point.

Ways to work on it

• Walkthrough. The Jacobian as the total derivative, an entry, and its determinant.
• Practice. Read off a Jacobian entry or determinant of a planar map.
• Hardest. Compute the Jacobian determinant of a change of coordinates and find where it fails to invert.

Not sure where to start? Take the ten-question placement test.