Calculate integral of Gaussian Geometry

Question

I have this excercise but I am having problems becouse I dont know how to use the Gauss-Bonnet theorem. If $r \in \mathbb{R}^{+}$ and $\Sigma_r$ is given by: $$\Sigma _ { r } = \left\{ ( x , y , z ) \in \mathbb { R } ^ { 3 } | z = \cos \sqrt { x ^ { 2 } + y ^ { 2 } } , x ^ { 2 } + y ^ { 2 } < r ^ { 2 } , x , y > 0 \right\} $$ Determinate the value of: $$\int_{\Sigma_r}KdA $$ If $K$ is the Gaussian Curvature of $\Sigma_r$. Help with this please.

Answer 1

Local Gauss-Bonnet Theorem. Let $R$ be a simple region of a surface $S$ and let $\alpha\colon I \to S$ be the boundary of the region. Assume $\alpha$ is positively oriented, parametrised by arc length $s$ and let $\theta_1, \ldots, \theta_k$ be the external angles at the vertices of $\alpha$. Then $$ \int_{R} K\, dA + \sum_{i=1}^k \int_{a_i}^{b_i} k_g(s)\,ds + \sum_{i=1}^k \theta_i= 2\pi,$$ where $K$ is the Gauss curvature and $k_g$ stands for the geodesic curvature of the regulars arcs of $\alpha$. (Reference: Curves and Surfaces from Do Carmo.)

So in order to calculate the integral, we need to calculate two things: (1) the total geodesic curvature along all arcs, and (2) the external angles at the vertices.

Step 0: Parametrizing the arcs
A parametrisation of $\Sigma_r$ is $$ x\colon \left[0,\frac{\pi}{2}\right]\times[0,r]: (u,v)\mapsto \left(v\cos u, v \sin u, \cos v\right). $$ This surface is bounded by the arcs $$ \begin{align*} \alpha_1(t)&= x(0,t) = (t,0,\cos t), \qquad \text{$t\in[0,r]$} \\ \alpha_2(t)&= x(t,r) = (r\cos t, r \sin t, \cos r), \qquad\text{$t\in[0,\frac{\pi}{2}]$} \\ \alpha_3(t)&= x\left(\frac{\pi}{2},t\right) = (0,r-t,\cos(r-t)), \qquad \text{$t\in[0,r]$.} \\ \end{align*} $$

Step 1: Calculating the geodesic curvature
Big hint: the meridians of a surface of revolution are geodesics. Use this fact, it will save you 66% of the work. If you didn't see this fact in your course, prove it or look it up.

Step 2: Calculating the external angles
Note that the arcs are coordinate lines. It is easily seen that $x$ is an orthogonal parametrisation of $\Sigma_r$, so the external angles at every vertex is $\frac{\pi}{2}$.

I hope this answer is helpful for you.