- Thread starter
- #1

- Feb 29, 2012

- 342

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.