Jacobian determinant

Definition

Let $U \subseteq \mathbb{R}^n$ be open, let $f \colon U \to \mathbb{R}^n$, and let $a \in U$ be a point at which all first partial derivatives of $f$ exist. The Jacobian determinant of $f$ at $a$ is the determinant $$\det J_f(a) = \det \left( \frac{\partial f_i}{\partial x_j}(a) \right)_{1 \le i, j \le n}$$ of the Jacobian matrix of $f$ at $a$. It is also written $\dfrac{\partial (f_1, \dots, f_n)}{\partial (x_1, \dots, x_n)}(a)$, and it is often called the Jacobian of $f$.

Remarks

For the polar map $(r, \theta) \mapsto (r \cos \theta, r \sin \theta)$, the Jacobian determinant is $r \cos^2 \theta + r \sin^2 \theta = r$. If $f$ is of class $C^1$ and $\det J_f(a) \ne 0$, then $f$ has a differentiable local inverse near $a$, by the inverse function theorem. The absolute value $|\det J_f|$ is the factor in the change of variables formula for multiple integrals.

Explanations

L1 human

A linear map of the plane sends the unit square to a parallelogram, and the determinant is the signed area of that parallelogram. The Jacobian determinant does the same for a curved map, at the scale of tiny squares near one point.