Find f(x,y,z) so that grad f = F = <2x + y + z, x + 3y^2 + z, x + y>
or show no such f exists.

Answer:
f_x = 2x+y+z    => f = x^2 + xy + xz + theta(y,z)
f_y = x+3y^2+z  => f = xy + y^3 + yz + psi(x,z)
f_z = x+y       => f = xz + yz       + phi(x,y)

So guess f = x^2 + y^3 + xy + xz + yz and check

f_x = 2x + 0 + y + z = 0   it checks
f_y = 0 + 3y^2 +x + 0 + z  it checks
f_z = 0 + 0 + 0 + x + y    it checks