What happens to differentials like $dA$, $dV$, $ds$, and $dS$ under a change of variables?

Question

Motivating example

I have noticed that the derivation of the moment of inertia of a solid ellipsoid uses a change of variables to transform it into a unit sphere; then the unit sphere's MoI gets scaled by the Jacobian determinant.

I wondered if I could use a similar technique to get the perimeter of an ellipse. (My ultimate goal is to compute the moment of inertia of a hollow ellipsoid, and I reason that if I can do an arc integral along an ellipse, then I can introduce another parameter to do a surface integral over the ellipsoid.)

The ellipse is defined by $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$

and its arc length is $$\int_C ds$$

where $C$ is the ellipse itself.

Now, the ellipse can be transformed into the unit circle if we assert

$$x = au \\ y = vb \\ \implies\\ u^2 + v^2 = 1$$

This transformation has

$$\left\vert J\right\vert = \left\vert \begin{matrix} a & 0 \\ 0 & b \\ \end{matrix}\right\vert = ab$$

And under this transformation,

$$\int_C ds = \int_{C'} \left\vert J\right\vert \,ds' = ab \int_{C'} ds'$$

But $C'$ is just the transformed ellipse, which is the unit circle! So this is just $ab$ times the circumference of the unit circle, and the perimeter is simply $$2\pi a b$$

I know this is wrong, so I thought that I need to transform $ds$ as well. I tried

$$ds = \sqrt{dx^2 + dy^2} = \sqrt{(a\,du)^2 +(b\,dv)^2}$$

which doesn't really get me anywhere, as I expected, given that there isn't a simple formula for the perimeter of an ellipse.

• Nonetheless, is this kind of manipulation correct in principle?
• If so, why isn't it possible to derive a closed-form expression for the perimeter of an ellipse from this?
• How come transforming $ds$ doesn't work here, but transforming $dV$ using the Jacobian in the above ellipsoid example works just fine? Is it because the moment of inertia problem considers a hollow ellipsoid, which would be analogous to integrating over the area of an ellipse (not its perimeter)?
• The 3D analogue of the perimeter would be the surface area, and a surface integral over the ellipsoid is required to find the moment of inertia of a hollow ellipsoid—I assume this latter quantity has no closed-form expression, either?

My central question

• In general, what happens to "geometric" differentials like $dA$, $dV$, $ds$, and $dS$ under a change of variables?

Answer 1

The "Jacobian" is used when you change coordinates in a volume integral, where it represents the dilation/contraction factor of neighborhoods as you go from one system to another. When you have a lower-dimensional integral, like an arc-length in the plane, or a surface integral in space, the corresponding correction factors under reparametrization are not the Jacobian. Say, if you have a curve $t\to\gamma(t)$ in the plane and you reparametrize $s\to\delta(s)$ then the "one-dimensional" Jacobian is just $|dt/ds|$, which is the ratio of the lengths of the tangent vectors. In the case of a surface in $\mathbb{R}^3$, the role of the Jacobian is played by ratio of the square roots of the determinants of the first fundamental forms, $\sqrt{EG-F^2}$ and $\sqrt{E'G'-F'^2}$ in either system of coordinates on the surface. This, again, represents the ratio of the surface area of the corresponding two-dimensional neighborhoods.

The first computation you did with the ellipse does not work. The second, however, is correct but you failed to relate $du$ and $dv$, as they are related on the ellipse (just write the implicit form of the ellipse in those coordinates and find the differential). The final integral should be one-dimensional, you end up integrating w.r.t. $u$ or $v$ or any other parameter on the ellipse.