What Are the Curvatures of the Quadric Surface at the Origin?

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Happy New Year everyone! Here is this week's POTW:

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Calculate the principal, Gaussian, and mean curvatures of the quadric surface

$$z = 2x^2 - xy + 3y^2$$

at the origin.-----

Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!

 

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No one answered this week's problem. You can read my solution below.

Spoiler

Writing

$$z = \frac{1}{2}\begin{bmatrix} x & y\end{bmatrix} \begin{bmatrix} 4 & -1\\-1 & 6\end{bmatrix} \begin{bmatrix}x\\ y \end{bmatrix}$$

we can compute the principal curvatures at the origin by finding the eigenvalues of the matrix

\begin{bmatrix}4 & -1\\-1 & 6\end{bmatrix}

Its characteristic polynomial is $t^2 - 10t + 23 = (t - 5)^2 - 2$, so the principal curvatures (i.e., the eigenvalues) are $\kappa_1 = 5+\sqrt{2}$ and $\kappa_2 = 5-\sqrt{2}$.

The Gaussian curvature is then $\kappa_1 \kappa_2 = 23$, and the mean curvature is $\frac{\kappa_1 + \kappa_2}{2} = 5$.