Question: a. Compute the gradient in polar coordinates. Note that it is not \(\left\langle\frac{\partial}{\partial r}, \frac{\partial}{\partial \theta}\right\rangle .\) In other words, compute the form so that the gradient vector is expressed in normal rectangular coordinates but evaluated at points in polar coordinates.

b. Check this by computing the gradient of \(f(x, y)=x^{2}+y^{2}\) and \(g(x, y)=\sin \left(x^{3}+y^{2}\right)\) in both rectangular coordinates and polar coordinates. Make sure to indicate how they are the same.

c. Compute the Laplacian \((\Delta f=\nabla \cdot \nabla f)\) in polar coordinates. It should come out to \(\Delta f=\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial f}{\partial r}\right)+\) \(\frac{1}{r^{2}} \frac{\partial^{2} f}{\partial \theta^{2}}+\frac{\partial^{2} f}{\partial z^{2}}\)

d. Check this by computing the Laplacian of \(f(x, y)=x^{2}+y^{2}\) and \(g(x, y)=\sin \left(x^{3}+y^{2}\right)\) in both rectangular coordinates and polar coordinates. Make sure to indicate how they are the same.

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