Let $\;U\subset \mathbb R^2\;$ be a smooth open domain and consider:
- $\;A\;$ a $\;2\times 2\;$ real matrix which depends on $\;x\;$, i.e $\;A=(a_{ij}(x))_{1\le i,j \le 2}\;$. $\;A\;$ is invariant under linear orthogonal transformation $\;x'=Ox\;$ where $\;O\;$ is a $\;2\times 2\;$ orthogonal matrix in the sense that $\;A'=OAO^T\;$ and invariant under translations (NOTE:$\;A'\;$ denotes matrix $\;A\;$ when depends on $\;x'\;$)
- $\;X=x_1\begin{pmatrix} a_{11}(x)\\a_{21}(x)\\ \end{pmatrix}+x_2\begin{pmatrix} a_{12}(x)\\a_{22}(x)\\ \end{pmatrix}\;$ for $\;(x_1,x_2) \in \mathbb R^2\;$ is a vector field such that $\;X=Ax\;$
- Fix $\;x_0 \in \partial U\;$ and consider the positively oriented orthonormal basis $\;\{v^{x_0},τ^{x_0}\}\;$ where $\;v^{x_0}\;$ is the outer normal vector and $\;τ^{x_0}\;$ is the tangential vector. Drop the superscript $\;x_0\;$
- The inner product $\;\langle \;,\;\rangle\;$ is with respect to the standard basis of $\;\mathbb R^2\;$ which is denoted by $\;\{x_1,x_2\}\;$
I want to apply divergence theorem on $\;X\;$ but I 'm having trouble dealing with the integral on the boundary of $\;U\;$. To be more specific, first I take the inner product $\;\langle X,v\rangle\;$ and I write:
$\;\langle X,v\rangle=\;\langle Ax,v\rangle=\;\langle OAx,Ov\rangle\;=\langle OAO^Tx',Ov\rangle=\;\langle A'x',Ov\rangle\;$
Then it should be $\;\int_{\partial U} \langle X,v\rangle\;dS=\int_{\partial U} \langle A'x',Ov\rangle\;dS\;$
QUESTIONS:
If I understand it right, since the inner product is with respect to the standard basis of $\;\mathbb R^2\;$ , $\;v\;$ is represented in the $\;\{x_1,x_2\}\;$ coordinate system as below (due to translation invariance of vectors):
$\;Ov\;$ is the rotated ,by some angle $\;\theta\;$, vector $\;v\;$ and $\;x'=(\langle x,τ\rangle, \langle x,v\rangle )\;$ are the coordinates of an arbitary $\;x\in \mathbb R^2\;$ with respect to $\;\{v,τ\}\;$ after rotation $\;x=O^Tx'\;$
If the above is true, then how can I apply divergence theorem? $\;Ov\;$ is NOT on boundary of $\;U\;$ but in the coordinate system $\;\{x_1,x_2\}\;$. Should I translate it back to $\;\hat x\;$ and how?
I'm having a really hard time getting my head around these two different coordinate systems. I don't understand where the rotation is happening and when I should translate the standard system to the new one.
I apologize for the long post and the bad drawing but I wanted to be as specific as possible. I 've been stuck so any help would be valuable.
Thanks in advance!