# Vector Potential

1. Homework Statement
Compute the vector potentials for the following two-dimensional fields:

$$\vec{F}$$ = $$\frac{xy\vec{i}}{1}$$ - $$\frac{y^{2}\vec{j}}{2}$$

2. Homework Equations
(1)$$\int^{1}_{0} t \vec{F}\times \frac{d\vec{r}}{dt} dt$$

3. The Attempt at a Solution
I solved this problem a different way, but my problem is using the above formula. The way that i did it was as follows:

$$\vec{k} \times$$$$\vec{F}$$ = $$\nabla\chi$$ = $$\left\langle\frac{y^{2}}{2},xy,0\right\rangle$$

so $$\chi$$ = $$\frac{xy^{2}}{2}$$

but, $$\vec{G} = \vec{k}\chi=\frac{xy^{2}}{2}\vec{k}$$
where $$\vec{F} = \nabla \times \vec{G}$$

This is the correct answer given by the book, with G being the vector potential. However, I cannot get to this answer using the formula (1) that I gave above. I did the following:

$$\vec{r}=t\left\langle x,y,z\right\rangle$$
$$\frac{d\vec{r}}{dt}=\left\langle x,y,z\right\rangle$$
$$\vec{F}(\vec{r}(t)) = \left\langle xyt^{2},\frac{t^{2}y^{2}}{2},0\right\rangle$$

Then, following through with the computations I get an incorrect answer. Did i set this up incorrectly? Any help at all would be greatly appreciated