As part of a longer definition I came across the following: $f: X\subseteq\mathbb{R}^m \rightarrow \mathbb{R}^n$ ($X$ open, $m
Intuitive explanation for Jacobian matrix having max. rank
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2 Answers
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$Df_x$ is an $m\times n$ matrix at each $x$, its maximal rank is $m$ (since $m
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.
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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.
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0But the matrix isn't square, so I don't see how this applies here. – 2012-05-30