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) = (y2 − 6x)i 2xyj

Answer 1

It looks like the vector field is

$\vec f(x,y) = (y^2-6x) \, \vec\imath + 2xy \, \vec\jmath$

$\vec f$ is conservative if we can find a scalar function $f(x,y)$ whose gradient is $\vec f$. This entails solving the partial differential equations

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

$\dfrac{\partial f}{\partial y} = 2xy$

Integrate both sides of the second PDE with respect to y :

$\displaystyle \int \frac{\partial f}{\partial y} \, dy = \int 2xy \, dy \implies f(x,y) = xy^2 + g(x)$

Differentiate with respect to x and solve for $g(x)$ :

$\dfrac{\partial f}{\partial x} = y^2 - 6x = y^2 + \dfrac{dg}{dx} \implies \frac{dg}{dx} = -6x \implies g(x) = -3x^2+C$

It follows that $\vec f$ is indeed conservative with potential function

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