I'm striving for a general explanation of integral transformation. So far I've been told some variable substitutions (like polar coordinates) without really getting the gist of it. However I've just got to know a method that makes the whole process pretty clear to me due to matrix multiplication. That method I want to generalize:
So according to the the rotation transformation above I ask myself whether there exist something alike:
$\left(\begin{array}{cc}x\\y\end{array}\right) = \Phi \cdot \left(\begin{array}{cc}u\\v\end{array}\right)$ in order that $x = u\,\cos(v)\quad y = u\,\sin(v)$. Could it be?
Best Answer
A matrix represents a linear transformation. And a linear transformation maps lines onto lines. Hence no linear transformation will have the properties you ask for.
You need to look at other kind of transformations!