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) = (2x − 6y) i + (−6x + 6y − 7) j

Answer 1

We're looking for a scalar function $f(x,y)$ such that $\nabla f(x,y)=\vec F(x,y)$. That is,

$\dfrac{\partial f}{\partial x}=2x-6y$

$\dfrac{\partial f}{\partial y}=-6x+6y-7$

Integrate the first equation with respect to $x$:

$f(x,y)=x^2-6xy+g(y)$

Differentiate with respect to $y$:

$-6x+6y-7=-6x+\dfrac{\mathrm dg}{\mathrm dy}\implies\dfrac{\mathrm dg}{\mathrm dy}=6y-7$

Integrate with respect to $y$:

$g(y)=3y^2-7y+C$

So $\vec F$ is indeed conservative with the scalar potential function

$f(x,y)=x^2-6xy+3y^2-7y+C$

where $C$ is an arbitrary constant.