Question

Determine whether or not f is a gradient vector field. if it is, find a function φ such that f = ∇φ. (if the vector field is not a gradient field, enter dne.) f(x, y) = (2x − 6y) i + (−6x + 10y − 7) j

Answer 1

$\mathbf f(x,y)=(2x-6y)\,\mathbf i+(-6x+10y-7)\,\mathbf j$

For $\mathbf f$ to be the gradient of some scalar function $\varphi(x,y)$, we need to find $\varphi$ that satisfies

$\dfrac{\partial \varphi}{\partial x}=2x-6y$
$\dfrac{\partial \varphi}{\partial y}=-6x+10y-7$

From the first PDE, it follows that (upon integrating with respect to $x$)

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

Differentiating with respect to $y$ yields

$\dfrac{\partial\varphi}{\partial y}=-6x+10y-7=-6x+\dfrac{\mathrm dg}{\mathrm dy}$
$\implies\dfrac{\mathrm dg}{\mathrm dy}=10y-7\implies g(y)=5y^2-7y+C$

$\implies\varphi(x,y)=x^2-6xy+5y^2-7y+C$