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Mean and Gaussian curvatures

Fantini

"Read Euler, read Euler." - Laplace
MHB Math Helper
Feb 29, 2012
342
The question is as follows:

Find the first and second fundamental forms, the normal vector and mean, Gaussian and principal curvatures of the surface given by $\varphi(u,v) = (u^n, v^n, uv).$

To calculate the first and second fundamental forms I did $\varphi_u = (n u^{n-1}, 0, v)$ and $\varphi_v = (0,n v^{n-1}, u)$ and from there follows that

$$\begin{cases}
E = \varphi_u \cdot \varphi_u = n^2 u^{2(n-1)} + v^2, \\
F = \varphi_u \cdot \varphi_v = uv, \\
G = \varphi_v \cdot \varphi_v = n^2 v^{2(n-1)} + u^2,
\end{cases}$$

whereas the normal can be found by doing $N = \varphi_u \times \varphi_v / \parallel \varphi_u \times \varphi_v \parallel$. My computations yield

$$N = \frac{1}{(v^{2n} + u^{2n})^{\frac{1}{2}}} (v^n, -u^n,0),$$

which can be used to compute the second fundamental form by doing

$$\begin{cases}
e = N \cdot \varphi_{uu} = n^2(n-1)u^{n-2}v^n, \\
f = N \cdot \varphi_{uv} = 0, \\
g = N \cdot \varphi_{vv} = -n^2(n-1)v^{n-2}u^n,
\end{cases}$$

but now I am lost as to how to find the mean, Gaussian and principal curvatures. I know the mean and Gaussian are $- \frac{1}{2} \text{tr } dN_p$ and $\det dN_p$ respectively, however I have no clue how to proceed.

Thanks. (Nod)