@TedShifrin I want to show that, if p, q and r are distinct positive numbers, there are exactly four umbilics on the ellipsoid $\frac{x^2}{p^2}+\frac{y^2}{q^2}+\frac{z^2}{r^2}=1$.
To show that I considered the parametrization $\sigma (u,v)=(p\cos u\sin v, q\sin u\sin v, r\cos v)$.
Then the principal curvatures are the roots of
$\begin{vmatrix}
L-\kappa E & M-\kappa F\\
M-\kappa F & N-\kappa G
\end{vmatrix}=0$.
Am I on the right way?
Today I was trying to solve an integral for a Fourier series. I looked at the solution and this was the solution:
\begin{align*}
C_n
&= \frac{1}{2\pi} \int_0^{2\pi} x^2 e^{-inx} \,\text{d}x \\
\xrightarrow{\text{integration by parts}}
C_n
&= \frac{1}{2\pi}
\left[
- \frac{x^2 e^{-inx}}{in}...
