(915, #2) Determine whether or not is a conservative vector field

(915, #5) Determine whether or not F(x,y) = < 1 + 4x³ y³, 3 x⁴ y²> is a conservative vector field. If it is, find a potential.

Solution: Write F = < f_x , f_y> for some f. Then:

Therefore the vector field is conservative.

To find the potential, f, we procede as follows:

We integrate f_xwith respect to x.

We then differentiate f with respect to y.

We compare this to the given f_y find that g'(y) = 0, so g(y) = k. That is

f(x,y) = x⁴ y³ + x + k

by C.A.H.