MATH 251 Lecture 15

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Derivatives of Polar Coordinates

${\begin{aligned}\mathrm {d} r&=r_{x}\mathrm {d} x+r_{y}\mathrm {d} y\\\mathrm {d} \theta &=\theta _{x}\mathrm {d} x+\theta _{y}\mathrm {d} y\end{aligned}}$

${\begin{pmatrix}\mathrm {d} x\\\mathrm {d} y\end{pmatrix}}={\begin{pmatrix}\cos {\theta }&-r\sin {\theta }\\\sin {\theta }&r\cos {\theta }\end{pmatrix}}{\begin{pmatrix}\mathrm {d} r\\\mathrm {d} \theta \end{pmatrix}}$

Inverse Matrix

${\begin{aligned}A&={\begin{pmatrix}a&b\\c&d\end{pmatrix}}\\A^{-1}&={\frac {1}{\mathrm {det} {\left(A\right)}}}{\begin{pmatrix}d&-b\\-c&a\end{pmatrix}}\end{aligned}}$

Solution

${\begin{pmatrix}\cos {\theta }&-r\sin {\theta }\\\sin {\theta }&r\cos {\theta }\end{pmatrix}}^{-1}={\begin{pmatrix}\cos {\theta }&\sin {\theta }\\-{\frac {\sin {\theta }}{r}}&{\frac {\cos {\theta }}{r}}\end{pmatrix}}$