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Consider a function $w=f(x,y)$, where $x=rcos\theta$ and $y=rsin\theta$. Verify each of the following:

$$ \frac{\partial{w}}{\partial{x}} = \frac{\partial{w}}{\partial{r}}\cos\theta - \frac{\partial{w}}{\partial{\theta}}\frac{\sin\theta}{r} $$

$$ \frac{\partial{w}}{\partial{y}} = \frac{\partial{w}}{\partial{r}}\sin\theta + \frac{\partial{w}}{\partial{\theta}}\frac{\cos\theta}{r} $$

$$ \left(\frac{\partial{w}}{\partial{x}}\right)^2 + \left(\frac{\partial{w}}{\partial{y}}\right)^2 = \left(\frac{\partial{w}}{\partial{r}}\right)^2 + \left(\frac{1}{r^2}\right) \left(\frac{\partial{w}}{\partial{\theta}}\right)^2 $$

I believe I am supposed to utilize the chain rule:

$$ Z = f(x(t), y(t)) $$ $$ \frac{\partial{Z}}{\partial{t}} = \frac{\partial{Z}}{\partial{x}} \frac{\partial{x}}{\partial{t}} + \frac{\partial{Z}}{\partial{y}} \frac{\partial{y}}{\partial{t}}, $$

but I am unsure how to modify the formula to obtain $\frac{\partial{Z}}{\partial{x}}$.

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1 Answer 1

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Using the chain rule, write $$ \begin{bmatrix} \frac{\partial w}{\partial r} \\ \frac{\partial w}{\partial \theta} \end{bmatrix} = \begin{bmatrix} \frac{\partial x}{\partial r} & \frac{\partial y}{\partial r} \\ \frac{\partial x}{\partial \theta} & \frac{\partial y}{\partial \theta} \end{bmatrix} \begin{bmatrix} \frac{\partial w}{\partial x} \\ \frac{\partial w}{\partial y} \end{bmatrix} =: \mathbf{J} \begin{bmatrix} \frac{\partial w}{\partial x} \\ \frac{\partial w}{\partial y} \end{bmatrix} $$ Compute $\mathbf{J}$: $$ \mathbf{J} = \begin{bmatrix} \cos\theta & \sin\theta \\ -r\sin\theta & r\cos\theta \end{bmatrix} $$ Find the inverse of $\mathbf{J}$ (which does not exist at $r = 0$): $$ \mathbf{J}^{-1} = \frac{1}{r}\begin{bmatrix} r\cos\theta & -\sin\theta \\ r\sin\theta & \cos\theta \end{bmatrix} $$ Then $$ \begin{bmatrix} \frac{\partial w}{\partial x} \\ \frac{\partial w}{\partial y} \end{bmatrix} = \mathbf{J}^{-1} \begin{bmatrix} \frac{\partial w}{\partial r} \\ \frac{\partial w}{\partial \theta} \end{bmatrix} = \frac{1}{r} \begin{bmatrix} r\cos\theta \frac{\partial w}{\partial r} - \sin\theta \frac{\partial w}{\partial \theta} \\ r\sin\theta \frac{\partial w}{\partial r} + \cos\theta \frac{\partial w}{\partial \theta} \end{bmatrix} $$ The expressions you need follow directly.

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