[tex]\mathbf f(x,y)=(2x-6y)\,\mathbf i+(-6x+10y-7)\,\mathbf j[/tex]
For [tex]\mathbf f[/tex] to be the gradient of some scalar function [tex]\varphi(x,y)[/tex], we need to find [tex]\varphi[/tex] that satisfies
[tex]\dfrac{\partial \varphi}{\partial x}=2x-6y[/tex]
[tex]\dfrac{\partial \varphi}{\partial y}=-6x+10y-7[/tex]
From the first PDE, it follows that (upon integrating with respect to [tex]x[/tex])
[tex]\varphi(x,y)=x^2-6xy+g(y)[/tex]
Differentiating with respect to [tex]y[/tex] yields
[tex]\dfrac{\partial\varphi}{\partial y}=-6x+10y-7=-6x+\dfrac{\mathrm dg}{\mathrm dy}[/tex]
[tex]\implies\dfrac{\mathrm dg}{\mathrm dy}=10y-7\implies g(y)=5y^2-7y+C[/tex]
[tex]\implies\varphi(x,y)=x^2-6xy+5y^2-7y+C[/tex]
