Homework 4

#exercise-problems #math

Problem Statement

Show the following facts:

$H = \frac{G L - 2 F M + E N}{2 (E G - F^{2})}$ , $K = \frac{L N - M^{2}}{E G - F^{2}}$
Principle curvature are roots of the equation:

k^{2} - 2 H k + K = 0

$k_{1} + k_{2} = 2 H$

Recall that $K = d e t (\begin{matrix} a_{1}^{1} & a_{2}^{1} \\ a_{1}^{2} & a_{2}^{2} \end{matrix})$ and $H = \frac{1}{2} t r (\begin{matrix} a_{1}^{1} & a_{2}^{1} \\ a_{1}^{2} & a_{2}^{2} \end{matrix})$ , where

\begin{aligned} (\begin{array}{c} a_{1}^{1} & a_{2}^{1} \\ a_{1}^{2} & a_{2}^{2} \end{array}) & = {(\begin{array}{c} E & F \\ F & G \end{array})}^{- 1} (\begin{array}{c} L & M \\ M & N \end{array}) \\ = \frac{1}{E G - F^{2}} (\begin{array}{c} G L - F M & G M - F N \\ E M - F L & E M - F M \end{array}) \end{aligned}

Thus indeed $K = d e t ({(\begin{matrix} E & F \\ F & G \end{matrix})}^{- 1} (\begin{matrix} L & M \\ M & N \end{matrix})) = \frac{L N - M^{2}}{E G - F^{2}}$ and $H = \frac{1}{2} t r (\frac{1}{E G - F^{2}} (\begin{matrix} G L - F M & G M - F N \\ E M - F L & E M - F M \end{matrix})) = \frac{G L - 2 F M + E N}{2 (E G - F^{2})}$ .

Let $k_{1}$ and $k_{1}$ be the principle curvatures which are the eigenvalues of the above matrix, then it is known that the trace of the matrix is equal to the sum of the eigenvalues so it immediately follows that $2 H = k_{1} + k_{2}$ .

To show that they satisfy the above equation we know that since $k_{i}$ are eigenvalues then they must satisfy the characteristic polynomial (in $k$ ) :

\begin{aligned} d e t ((\begin{array}{c} a_{1}^{1} & a_{2}^{1} \\ a_{1}^{2} & a_{2}^{2} \end{array}) - (\begin{array}{c} k & 0 \\ 0 & k \end{array})) & = 0 \\ ⟹ - \frac{G k L}{E G - F^{2}} + \frac{2 F k M}{E G - F^{2}} - \frac{E k N}{E G - F^{2}} - \frac{F^{2} L N}{{(E G - F^{2})}^{2}} + \frac{E G L N}{{(E G - F^{2})}^{2}} + \frac{F^{2} M^{2}}{{(E G - F^{2})}^{2}} - \frac{E G M^{2}}{{(E G - F^{2})}^{2}} + k^{2} & = 0 \\ ⟹ k^{2} - k (\frac{G L - 2 F N + E N}{E G - F^{2}}) + \frac{E G (L N - M^{2}) - F^{2} (L N - M^{2})}{(E G - F^{2})^{2}} & = 0 \\ ◻ & ⟹ k^{2} - 2 k H + K & = 0 \end{aligned}