$\underline{G}=(8r^7x-5r^3y)\underline{i}+(-8r^7y+5r^3x)\underline{j}$
where $r=(x^2+y^2)^\frac{1}{2}$
How would I express the vector field as cylindrical coordinates? I have looked at various examples but none of them translate to an example such as this, any help would be fantastic!
First of all you will want to use the substitutions $x =r\cos \theta$ and $y = r\sin \theta$ for the polar angle $\theta$. The next part is in changing the unit vectors from $\hat{i}$ and $\hat{j}$ to $\hat{r}$ and $\hat{\theta}$. Verify for yourself that $\hat{r} = \frac{1}{\sqrt{2}}( \hat{i} + \hat{j})$. We will want $\hat{\theta}$ to be orthogonal to $\hat{r}$ (think of this as pointing in the direction of increasing $\theta$) so this can be taken to be $\hat{\theta} = \frac{1}{\sqrt{2}}( \hat{i} - \hat{j})$. You can solve the equations for $\hat{i}$ and $\hat{j}$ in terms of $\hat{r}$ and $\hat{\theta}$ and substitute the values for $x$ and $y$ as well.