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) = (2x − 6y) i + (−6x + 6y − 7) j

We're looking for a scalar function [tex]f(x,y)[/tex] such that [tex]\nabla f(x,y)=\vec F(x,y)[/tex]. That is,[tex]\dfrac{\partial f}{\partial x}=2x-6y[/tex][tex]\dfrac{\partial f}{\partial y}=-6x+6y-7[/tex]Integrate the first equation with respect to [tex]x[/tex]:[tex]f(x,y)=x^2-6xy+g(y)[/tex]Differentiate with respect to [tex]y[/tex]:[tex]-6x+6y-7=-6x+\dfrac{\mathrm dg}{\mathrm dy}\implies\dfrac{\mathrm dg}{\mathrm dy}=6y-7[/tex]Integrate with respect to [tex]y[/tex]:[tex]g(y)=3y^2-7y+C[/tex]So [tex]\vec F[/tex] is indeed conservative with the scalar potential function[tex]f(x,y)=x^2-6xy+3y^2-7y+C[/tex]where [tex]C[/tex] is an arbitrary constant.