Definition · applies to functions with as many outputs as inputs · also called Jacobian, $\det Df(x)$, $\frac{\partial(y_1, \dots, y_n)}{\partial(x_1, \dots, x_n)}$

High school

Consider a function $f(x, y)$ that maps each point of the $xy$-plane to a new point in the plane. Near a given point, $f$ stretches or shrinks tiny regions, and it may also flip them over, like a mirror. The Jacobian determinant of $f$ at that point is the factor by which $f$ scales the areas of those tiny regions, made negative if $f$ flips them over.

Undergraduate

Consider a differentiable function $f : \reals^n \to \reals^n$ (which has as the same number of outputs as inputs, so its Jacobian matrix $Df(x)$ at each point $x$ is square). The Jacobian determinant of $f$ at $x$ is the determinant of that matrix, $\det Df(x)$.

Properties

  • Change of variables: if $f$ is a one-to-one, continuously differentiable map from a region $U$ onto a region $V$, then \[\int_V g(y) \mathrm{d}y = \int_U g(f(x)) \lvert \det Df(x) \rvert \mathrm{d}x.\]
  • Inverse function theorem: if $\det Df(x) \neq 0$ and $f$ is continuously differentiable, then $f$ has a continuously differentiable inverse near $x$.
  • The Jacobian determinant of a composition is the product of the Jacobian determinants: $\det D(g \circ f)(x) = \det Dg(f(x)) \cdot \det Df(x)$.

Notes

It is often called simply “the Jacobian,” which can also mean the Jacobian matrix. See Jacobian: matrix vs. determinant.

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