Integral of mean curvature function

[neelakash]

Hello everyone,
I am self teaching some elementary notions of differential geometry. Rather, I should say I am concentrating on mean and gaussian curvature concepts related to a physics application I am interested in. I see one has to evaluate an integral that goes as:

$$\int_{\partial\Omega}\kappa\hat{n}\cdot d\vec{r}$$

-where $$\hat{n}$$ denotes the unit normal vector to the surface $$\partial\Omega$$. The integral is done over all the points $$\vec{r}$$ on the surface. Can anyone tell if there is a prescription to evaluate such integrals as this? If some reference is cited, that will also be helpful.

-Neel

 

[quasar987]

You mean "how do we compute this?" ??

If the surface has some nice symmetry spherical or cylindrical symmetry like if it's a conic (ellipsoid, paraboloid, wheteveroid), then you can use the relevant coordinate system (spherical or cylindrical) to integrate.

 

[neelakash]

Well, of course you are right...But, what I was expecting is a bit different...I was wondering if this integral could somehow be related to Gaussian curvature. The physics motivation is that for some specific type of surfaces, the integral which apparently may also be written as:
$$\int\nabla_{LB}\vec{r}(u,v)\cdot d\vec{r}(u,v)$$

should contain Gaussian curvature. Here $$\nabla_{LB}$$ is the Laplace-Beltrami operator.

 

[lavinia]

I think you need to extend the mean curvature of the surface to the interior of the volume that it bounds. Otherwise I am not sure how you would use Stokes theorem.

 

[blue_raver22]

Hello Neel,

Thank you for your question. The integral you have mentioned is known as the integral of mean curvature function. It is a fundamental concept in the field of differential geometry and is often used in physics applications as well. To evaluate this integral, one can use the Gauss-Bonnet theorem, which states that the integral of the mean curvature function over a closed surface is equal to the Euler characteristic of that surface.

In order to evaluate the integral, you will first need to determine the mean curvature function, which is given by \kappa = \frac{1}{2H}, where H is the mean curvature. Once you have the mean curvature function, you can then use the Gauss-Bonnet theorem to evaluate the integral over the surface \partial\Omega. You can refer to any standard textbook on differential geometry for more information on the Gauss-Bonnet theorem and its applications.

I hope this helps. Best of luck with your studies in differential geometry!