Intuitive explanation for Jacobian matrix having max. rank

Question

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.

Answer 1

$Df_x$ 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.

Answer 2

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.