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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:

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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?

Leon
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    Here is a former answer of mine giving a solution in terms of a certain complex function connectd to elliptic integrals. – Jean Marie Apr 27 '21 at 10:22
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    With keywords sequence "mapping circle onto square" Google finds you a lot of references, some of them very simple. The advantage of conformal mapping is angle preservation. – Jean Marie Apr 27 '21 at 10:37

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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!

mathcounterexamples.net
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