How did they go from the first set of partials to the second set of partials? In other words, how did they factor out $\frac{df}{dx}$ and $\frac{df}{dy}$ ?
2026-03-31 10:06:51.1774951611
How did they get these partial derivatives?
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The first two equations are a system of equations $Au = v$, where $A = \begin{bmatrix} \cos \theta & \sin \theta \\ -r \sin \theta & r \cos \theta \end{bmatrix}$, $ u = \begin{bmatrix} \frac{\partial f}{\partial r} \\ \frac{\partial f}{\partial \theta}\end{bmatrix} $, $ v = \begin{bmatrix} \frac{\partial f}{\partial x} \\ \frac{\partial f}{\partial y}\end{bmatrix} $, and so the bottom equations are simply $ u = A^{-1}v$.