Question: Is there a general relationship between surface area and volume analogous to the below examples?
Example 1. Consider a ball $B$ centered at the origin of a spherical coordinate system. The ball has volume $V(r) = \frac{4}{3}\pi r^3$, where $r$ is the radial distance from the origin to the boundary $\partial B$ of the ball. Now consider the surface area of $\partial V$. Elementary geometry shows that $A(r) = 4 \pi r^2$. Comparing $A$ and $V$, it follows that
$$ \frac{\partial V}{\partial r} = A. $$
Example 2. Another simple example is an open cylinder of length $l$. Place the cylinder at the origin and then define $s$ as the radial distance from the origin (in cylindrical coordinates). The volume of the cylinder is $V(s) = \pi s^2 l$ and its surface area is $A(s) = 2 \pi s l$. Once again, we have a relationship between volume and surface area with reference to the variable of variation (e.g., $s$ is the variable of variation, since both the volume and cylinder share the common length $l$),
$$ \frac{\partial V}{\partial s} = A. $$
Example 3. Although a cube may seem to stray from this relationship, as $\partial_s (s^3) \neq 6s^2$, this is only due to the lack of symmetry. If we center the cube in a Cartesian coordinate system, and define $a$ to be half the side length $s$, we see that $V(a) = (2a)^3 = 8a^3$ and $A(a) = 6(2a)^2 = 24a^2$. Thus,
$$ \frac{\partial V}{\partial a} = A. $$
Now consider a region $Q \subset \mathbb{R}^3$ with boundary $\partial Q$. The "center of mass" is defined as
$$ \textbf{r}_{cm} = \frac{1}{V(Q)} \int_{Q} \textbf{r} dV, $$
where $\textbf{r} = (x, y, z)$ in Cartesian coordinates. Assume that the object is centered in a coordinate system such that $\textbf{r}_{cm} = \textbf{0}$. Now let a coordinate system be described by a basis $\{\textbf{x}_i\}$, where $i = 1, 2, 3$. In general, when will the following relationship hold between $V = V(Q)$ and $A(\partial Q)$,
$$ \frac{\partial V}{\partial x_i} = A? $$
Also, assuming this relationship holds in some coordinate system, is there only one associated region that satisfies the necessary conditions (e.g., a sphere in spherical coordinates, a cube in Cartesian coordinates)?
