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For each of the following, either find the potential function or state that the vector field is not conservative.

1.  \(\vec{F}(x, y) = \langle ye^{xy},xe^{xy} \rangle\)

a.  \(f(x,y) = xy e^{xy}\)

b.  \(f(x,y) = e^{xy}\)

c.  \(f(x,y) = (y + x) e^{xy}\)

d.  \(f(x,y) = \frac{x^2 + y^2} e^{xy}\)

e.  \(\vec{F}\) is not conservative.

2.  \(\vec{F}(x, y) = \langle 2xy + y^2, x^2 + 2xy \rangle\)

a.  \(f(x,y) = x^2 y + x y^2\)

b.  \(f(x,y) = 2xy + x^2 + y^2\)

c.  \(f(x,y) = 2xy + \frac{1}{3}x^3 + \frac{1}{3}y^3\)

d.  \(f(x,y) = x^2y^2 + \frac{1}{3}x^3 + \frac{1}{3}y^3\)

e.  \(\vec{F}\) is not conservative.

3.  \(\vec{F}(x, y) = \langle xy + y^2, x^2 + xy \rangle\)

a.  \(f(x,y) = x^2 y + x y^2\)

b.  \(f(x,y) = 2xy + x^2 + y^2\)

c.  \(f(x,y) = 2xy + \frac{1}{3}x^3 + \frac{1}{3}y^3\)

d.  \(f(x,y) = x^2y^2 + \frac{1}{3}x^3 + \frac{1}{3}y^3\)

e.  \(\vec{F}\) is not conservative.