Inverse Function Theorem

A nonzero Jacobian determinant certifies a local inverse, and the inverse's derivative is the matrix inverse.

The idea

Theorem (Inverse function theorem).

Let $f \colon \mathbb{R}^{n} \to \mathbb{R}^{n}$ be $C^{1}$ on an open set containing $a$, and suppose the derivative matrix $Df(a)$ is invertible, equivalently $\det Df(a) \ne 0$. Then there are open sets $U$ containing $a$ and $V$ containing $f(a)$ such that $f$ maps $U$ one-to-one onto $V$, the inverse map $f^{-1} \colon V \to U$ is again $C^{1}$, and $D(f^{-1})\bigl(f(a)\bigr) = \bigl[Df(a)\bigr]^{-1}.$

The theorem transfers a property of one linear map, checked at a single point, to the nonlinear map on a whole neighbourhood: the derivative $Df(a)$ is the best linear approximation of $f$ at $a$, and if that linear map can be undone, so can $f$ near $a$.

The conclusion is local. A map can have invertible derivative at every point of its domain and still fail to be one-to-one globally, by wrapping the domain around itself; the theorem promises an inverse near each point, never one inverse everywhere.

Ways to work on it

• Walkthrough. The hypothesis, a worked Jacobian, and the inverse's derivative.
• Practice. Decide local invertibility from a derivative matrix.
• Hardest. Decide where a change of coordinates inverts, and compute the inverse's Jacobian.

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