a. If [tex]\vec F[/tex] is a gradient field of a scalar function [tex]f[/tex], then
[tex]\dfrac{\partial f}{\partial x}=6xy^2[/tex]
[tex]\dfrac{\partial f}{\partial y}=6x^2y[/tex]
From these equations we can find
[tex]f(x,y)=3x^2y^2+g(y)[/tex]
[tex]\dfrac{\partial f}{\partial y}=6x^2y=6x^2y+\dfrac{\mathrm dg}{\mathrm dy}\implies\dfrac{\mathrm dg}{\mathrm dy}=0\implies g(y)=C[/tex]
[tex]\implies\boxed{f(x,y)=3x^2y^2+C}[/tex]
b. By the gradient theorem (i.e. fundamental theorem of calculus for line integrals),
[tex]\displaystyle\int_C\nabla f\cdot\mathrm dr=f(\vec r(1))-f(\vec r(0))=\boxed{12}[/tex]
