Regular point - Knowlpedia

A regular point of a differentiable map $F:U\to \mathbb{R}^m$ , where $U\subseteq\mathbb R^n$ is open, is a point $a\in U$ such that the Fréchet derivative $DF(a):\mathbb{R}^n\to \mathbb{R}^m$ is surjective.

Equivalent characterizations

Equivalently, the Jacobian matrix $JF(a)$ has rank $m$ . Thus regular points can exist only when $m\le n$ .

Remarks

Regular points are the local nondegeneracy condition used in the implicit function theorem and in the definition of a regular value.

Examples

For $F(x,y)=x^2+y^2$ , every point $(x,y)\ne(0,0)$ is regular because the gradient $(2x,2y)$ is nonzero.
For $F(x,y)=(x^2,y^2)$ , the point $(1,1)$ is regular, while $(0,1)$ is not because the Jacobian has rank $1$ there.