Can't wrap my head around a question I saw today here: https://math.stackexchange.com/questions/2121993/if-f-is-holomorphic-and-left-f-right-is-constant-then-f-is-constant#:~:text=If%20f%20is%20holomorphic%20and,constant%20then%20f%20is%20constant&text=Thus%20ux%3Duy,and%20thus%20f%20is%20constant.
Why can I say that the determinant of the Jacobian is indeed zero and not any constant? I can't understand that step.
Thanks in advance!
If $f$ is a vector function of a vector $x$, both in $\mathbb{R}^n$, such that
$ \| f \| = \text{Constant} $
Then
$ \nabla_x (f^T f) = 0 $
But
$ \nabla_x (f^T f) = 2 J^T f $
where $ J $ is the Jacobian of $f$ with respect to $x$.
Hence,
$ 2 J^T f = 0 $
Since $f$ is not necessarily the zero vector, then it follows that $J$ must be rank deficient, i.e. $| J | = 0 $.