Question

Determine whether or not F is a conservative vector field. If it is, find a function f such that F = ∇f. (If the vector field is not conservative, enter DNE.) F(x, y) = (3x2 − 2y2)i + (4xy + 4)j

Answer 1

In order for F to be conservative, there must be a scalar function f such that the gradient of f is equal to F. This means

$\dfrac{\partial f}{\partial x}=3x^2-2y^2$

$\dfrac{\partial f}{\partial y}=4xy+4$

Integrate both sides of the first equation with respect to x :

$f(x,y)=x^3-2xy^2+g(y)$

Differentiate both sides with respect to y :

$\dfrac{\partial f}{\partial y}=-4xy+\dfrac{\mathrm dg}{\mathrm dy}=4xy+4\implies\dfrac{\mathrm dg}{\mathrm dy}=8xy+4$

But we assume g is a function of y, which means its derivative can't possibly contain x, so there is no scalar function f whose gradient is F. Therefore F is not conservative.