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Let $E= \{\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2} \le 1\}$ be a solid ellipsoid. For every point $p \in \partial E$ we define $\rho(p)$ to be the distance between the origin and the affine tangent space at $p$.

Calculate the integrals $\displaystyle \int_{\partial E}\rho \, dS$ and $\displaystyle \int_{\partial E} \frac{1}{\rho} \, dS$.

I was able to calculate the first, I got $3\operatorname{Vol}(E)$. However, I have no idea about the second. To do that I found the perpendicular normal vector $n(p)$ and then $\rho(p) = \langle p,n(p) \rangle$ and used the divergence theorem.

Generally, the solution should be related to those subjects (divergence, flux).

Any tip or advice would be appreciated.

Thanks!

Ng Chung Tak
  • 19,693
Yotam Alon
  • 135

1 Answers1

2

Equation of tangent plane at $(x',y',z')$

$$\frac{x'x}{a^2}+\frac{y'y}{b^2}+\frac{z'z}{c^2}=1$$

$$\rho= \frac{1}{\sqrt{\dfrac{x'^2}{a^4}+\dfrac{y'^2}{b^4}+\dfrac{z'^2}{c^4}}}$$

Let $ \begin{pmatrix} x' \\ y' \\ z' \end{pmatrix}= \begin{pmatrix} a\sin u \cos v \\ b\sin u \sin v \\ c\cos u \end{pmatrix}$

\begin{align*} dS &= abc\sqrt{\dfrac{x'^2}{a^4}+\dfrac{y'^2}{b^4}+\dfrac{z'^2}{c^4}} \sin u \,du \, dv \\ \oint_{\partial E} \rho \, dS &= abc \int_{0}^{2\pi} dv \int_{0}^{\pi} \sin u\, du \\ &= 4\pi abc \end{align*}

For the second integral \begin{align*} \oint_{\partial E} \frac{1}{\rho} \, dS &= abc \int_{0}^{2\pi} \int_{0}^{\pi} \left( \frac{\sin^2 u \cos^2 v}{a^2}+ \frac{\sin^2 u \sin^2 v}{b^2}+ \frac{\cos^2 u}{a^2} \right) \sin u\, du \, dv \\ &= abc \int_{0}^{2\pi} \left( \frac{4\cos^2 v}{3a^2}+ \frac{4\sin^2 v}{3b^2}+ \frac{2}{3c^2} \right) dv \\ &= \frac{4\pi abc}{3} \left( \frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2} \right) \end{align*}

Ng Chung Tak
  • 19,693