Question: Let \(z=f(x, y)\) be a differentiable function. If we change to polar coordinates, we make the substitution \(x=r \cos (\theta), y=r \sin (\theta)\), where \(x^{2}+y^{2}=r^{2}\) and \(\tan (\theta)=\frac{y}{x}\).

1. Find expressions for \(\frac{\partial z}{\partial r}\) and \(\frac{\partial z}{\partial \theta}\) involving \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\).
2. Show that \(\left(\frac{\partial z}{\partial x}\right)^{2}+\left(\frac{\partial z}{\partial y}\right)^{2}=\left(\frac{\partial z}{\partial r}\right)^{2}+\frac{1}{r^{2}}\left(\frac{\partial z}{\partial \theta}\right)^{2}\).

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