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Vector Potential

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

[tex]\vec{F}[/tex] = [tex]\frac{xy\vec{i}}{1}[/tex] - [tex]\frac{y^{2}\vec{j}}{2}[/tex]


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


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:

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

so [tex]\chi[/tex] = [tex]\frac{xy^{2}}{2}[/tex]

but, [tex]\vec{G} = \vec{k}\chi=\frac{xy^{2}}{2}\vec{k}[/tex]
where [tex]\vec{F} = \nabla \times \vec{G}[/tex]

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:

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

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
 

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