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Vector potential of a 2 dimensional field

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1. Homework Statement
Compute vector potential for the following field:

[tex]\vec{F}[/tex] = <xy, y^2/2 >


2. Homework Equations

[tex]^{0}_{1}[/tex][tex]\int[/tex] t[tex]\vec{F}[/tex] x [tex]\frac{d\vec{r}}{dt}[/tex]dt


3. The Attempt at a Solution
I set r = tR, where R = <x,y,z>. Taking the crossproduct F x R, I get <0,0,xy[tex]^{2}[/tex] + [tex]\frac{xy^{2}}{2}[/tex]>. I then have an integral:

Int (t <0,0,xy[tex]^{2}[/tex] + [tex]\frac{xy^{2}}{2}[/tex]>)dt which gives me
<0,0,xy[tex]^{2}[/tex] + [tex]\frac{xy^{2}}{2}[/tex]>1/2

this simplifies to
<0,0,[tex]\frac{3xy^{2}}{4}[/tex]>, which is an incorrect answer ( the right being <0,0,xy^2/2>.

I honestly have no clue what I'm doing wrong. My book doesn't show any examples, and doesn't explain how they get this formula so It's been a plug&chug problem for me that hasn't worked. I tried to look online for a description of what is going on but I havent found anything. If anyone knows where I could find examples to similar problems online and maybe where I could find how to derive the formula i used, or could help me out I would appreciate it greatly.
 

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