# Intuitive explanation for Jacobian matrix having max. rank

As part of a longer definition I came across the following: $f: X\subseteq\mathbb{R}^m \rightarrow \mathbb{R}^n$ ($X$ open, $m<n$) with $rank(Df_{x}) = m$ for all $x \in X$. My question is now if you can give me an intuitive explanation of what it means for $f$ that its Jacobian matrix has rank $m$ for all $x\in W$. I don't know if it matters but $f$ was also to be injective and smooth.

## 2 Answers

$Df_x$ is an $m\times n$ matrix at each $x$, its maximal rank is $m$ (since $m<n$). $Df_x$ has maximal rank at every $x$ means that $f$ is an immersion.

http://en.wikipedia.org/wiki/Immersion_%28mathematics%29

Roughly speaking that means the tangent space at $x$ is mapped injectively into the tangent space at $f(x)$. That map between tangent spaces is called the differential map, represented by the Jacobian matrix in this case. Hope that helps.

The determinant of the Jacobian is the constant by which a volume is multiplied in a very small area around the point if you apply the function.

If the determinant is 0, this means that the function maps the point into a "smaller" space which does not go well with being injective.

• But the matrix isn't square, so I don't see how this applies here. – ghostclicker May 30 '12 at 15:17