I'm considering the problem of computing the volume and surface area of an ellipsoid. We define the ellipsoid by the set of points

$$S=\left\{(\alpha r\cos\theta\sin\phi,\beta r\cos\theta\cos\phi,\gamma r\sin\theta)\,:\,\,0\leq\phi\leq2\pi,0\leq\theta\leq\pi,0\leq r\leq\rho\right\}$$

where, naturally, $\alpha$, $\beta$, $\gamma$, and $\rho$ are positive constants.

If we'd like to compute the volume of the ellipsoid, can first make a change of variable from Cartesian coordinates $(x,y,z)$ to scaled Cartesian coordinates $(u,v,w)$ by a mapping $\Phi_1$:

$$(x,y,z)=\Phi_1(u,v,w)=(\alpha u,\beta v,\gamma w)$$

such that our volume integral is


We then make a second change of variable from our (scaled) Cartesian coordinates to spherical coordinates, giving

$$\begin{aligned}\text{vol}(S)&= \alpha\beta\gamma\int_{r=0}^{\rho}\int_{\theta=0}^{\pi}\int_{\phi=0}^{2\pi}r^2\sin\theta\,d\phi\,d\theta\,dr\\ &=\frac{4}{3}\pi\,\alpha\beta\gamma\rho^3\end{aligned}$$

which is certainly the correct formula.

I had the (rather reckless) idea of trying a similar thing with the integral for the surface area of the ellipsoid. We define $\Phi_1$ by $$\Phi_1(u,v,w)=(\alpha u,\beta v,\gamma w)$$ and $\Phi_2$ by $$\Phi_2(\phi,\theta)= (\cos\theta\sin\phi,\cos\theta\cos\phi,\sin\theta)$$ It's obvious that composing $\Phi_1$ with $\Phi_2$ gives the desired parametrization of the ellipsoid. However, if we try playing the same game is with the volume integral by making two successive changes of variable, we end up with $$\begin{aligned}\text{area}(S)&=\alpha\beta\gamma\rho^3\int_{\theta=0}^{\pi}\int_{\phi=0}^{2\pi}|\cos\theta|\,d\phi\,d\theta\\ &=4\pi\,\alpha\beta\gamma\rho^2\end{aligned}$$ which is certainly not the correct formula.

My question is regarding what, precisely, went wrong with the surface area computation. My best guess is that the mapping $\Phi_2$ is just a parametrization of the surface, and therefore doesn't have any reason to play nicely with changes of variable. However, I'm a little unclear on the specifics of what's happening. Can anyone offer any insight on the distinction between the bona fide change of variables in the volume integral and the mapping from 2-space to 3-space in the volume integral?


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