Continuum Mechanics Homework - Vector Field in Polar Coordinates

[Pooty]

Hi, so I scanned an image of the problem statement and my attempt at the solution. I don't know if I am headed in the right direction and need some guidance. This is my first post ever and I hope I am doing this properly. Thank you for any help you guys can provide.

 

[CompuChip]

I think you got the expression for the gradient wrong. Note that V should again be a vector, so you should have an expression like

$$\nabla f(r, \theta) = \frac{df}{dr} \hat e_r + \frac{1}{r} \frac{df}{d\theta} \hat e_\theta$$