from cartesian coordinates to polar coordinates

Question

The gradient is known as $\nabla u=(u_x,u_y)$ .

Let $x=r\cos(\phi),\ y=r\sin(\phi)$. Does it then become $(u_r,\frac{1}{r}u_\phi)$? I have given that $u(r\cos(\phi),r\sin(\phi))=r^3\cos(\phi)$. From this I have to calculate the gradient.

Answer 1

If you have a function in (non-spherical) polar coordinates i.e. $f(r, \phi)$, then the gradient is given as

$$\nabla f(r, \phi) =\dfrac{\partial f}{\partial r}\hat{r} + \dfrac{1}{r}\dfrac{\partial f}{\partial \phi}\hat{\phi}$$