While messing around with the notion of the adjoint operator in a euclidean space, I asked myself the following question:
Given $f:U\subseteq\mathbb{R}^n\rightarrow\mathbb{R}^n$ differentiable on $U$, for any $x\in U$, what can be said on $(\mathrm{d}_xf)^*$?
$(\mathrm{d}_xf)^*$ is the adjoint of $\mathrm{d}_xf\in\textrm{End}(\mathbb{R}^n)$ with respect of the canonical euclidean structure of $\mathbb{R}^n$.
Let $(e_1,\cdots,e_n)$ be the canonical basis of $\mathbb{R}^n$, $f=(f_1,\cdots,f_n)$ and $x=(x_1,\cdots,x_n)\in U$. For any $h=(h_1,\cdots,h_n)\in\mathbb{R}^n$, one has: $$(\mathrm{d}_xf)^*.h=\sum_{i=1}^n\sum_{j=1}^n\frac{\partial f_j}{\partial x_i}h_je_i.$$
Is there a neat (coordinate-wise independent) way to characterize/express $(\mathrm{d}_xf)^*$?
If $f$ is a submersion at every point above $0$ and $M=f^{-1}(0)$, one has: $$\textrm{im}(\mathrm{d}_xf)^*=(T_xM)^{\perp}.$$
In the same idea, is there some geometric interpretations of $(\mathrm{d}_xf)^*$?
Any enlightenment will be greatly appreciated.