Given a scalar function (field?) $f(\mathbf{x})$ (where $\mathbf{x}=[x_1,x_2,x_3]$) with gradient
$$\nabla f= [f_{x_1},f_{x_2},f_{x_3}]$$
Where
$$f_{x_i}=\frac{\partial f}{\partial x_i}$$
And given
$$\nabla f \cdot \mathbf{x}=c$$
where $c$ is some constant,
Is there a closed surface such that the outward pointing normal vector is everywhere equal to the argument vector $\mathbf{x}$, so that:
$$\int \int_S \nabla f \cdot \mathbf{x} \:dS=cA$$
where $A$ is the area of the closed surface. And can this closed surface, if it exists, be arbitrarily chosen such that $A=1$ or $A=4\pi r^2$ as convenient, or is it uniquely determined?