Jacobian Rank Local Behaviour Theorem¶

Theorem¶

Let $\boldsymbol{f}\colon U\subseteq\mathbb{R}^n\to\mathbb{R}^m$ be continuously differentiable near $\boldsymbol{a}$ , and write $\boldsymbol{D}\boldsymbol{f}(\boldsymbol{a})$ for its $m\times n$ Jacobian matrix at $\boldsymbol{a}$ .

If $\operatorname{rank}\boldsymbol{D}\boldsymbol{f}(\boldsymbol{a})=n$ and $n\le m$ $m$ , then $\boldsymbol{f}$ is locally injective near $\boldsymbol{a}$ .

If $\operatorname{rank}\boldsymbol{D}\boldsymbol{f}(\boldsymbol{a})=m$ and $m\le n$ $n$ , then $\boldsymbol{f}$ is locally surjective near $\boldsymbol{a}$ .

If $n=m$ and $\det \boldsymbol{D}\boldsymbol{f}(\boldsymbol{a})\neq 0$ $0$ , then $\boldsymbol{f}$ is locally bijective near $\boldsymbol{a}$ .

Interpretation¶

Full rank of the Jacobian tells which local behaviour survives under linearization: no collapsing for full column rank, no missing nearby outputs for full row rank, and a local inverse in the square case.

Two examples. Top: a map from the line into the plane with full rank keeps nearby inputs distinct (locally injective). Bottom: a map from a plane region onto an interval of the line covers all nearby values (locally surjective).

In the top example, $n=1\le m=2$ and $\operatorname{rank}\boldsymbol{D}\boldsymbol{f}=1$ $1$ . Distinct nearby inputs have distinct images, illustrating local injectivity.

In the bottom example, $m=1\le n=2$ and $\operatorname{rank}\boldsymbol{D}f=1$ $1$ . The map covers a whole interval of nearby target values, illustrating local surjectivity.

Local injectivity and surjectivity describe neighbourhoods, rather than the entire domain and codomain. Full column rank prevents nearby inputs from sharing an output; full row rank permits every sufficiently nearby output to be reached.

Key Results¶

The square case is the multivariable inverse function theorem.
Full column rank gives a locally one-to-one parametrized surface or curve.
Full row rank means nearby target values can be reached by nearby inputs.

Examples¶

For $\boldsymbol{f}(t)=(t,t^2)$ $(t, t^{2})$ , $\boldsymbol{D}\boldsymbol{f}(t)=(1,2t)^{\top}$ has rank $1$ , so the curve is locally injective.
For $f(x,y)=x+y$ $y$ , $\boldsymbol{D}f=(1\ 1)$ has rank $1$ , so the map is locally surjective onto $\mathbb{R}$ .

Prerequisites¶

Sources¶

Math IV: Multivariate Calculus · Week 3, Lecture 1

Aliases: Jacobian Rank Local Behavior Theorem Keywords: Jacobian rank theorem, full rank local injectivity, full rank local surjectivity Symbols: real numbers, subset

Last updated: 17 October 2026 21:53