Integrating Tensor on Riemannian 2-Manifold at Point p

[GPPaille]

I have a riemannian 2-manifold (Let's say a sphere) and a tensor of order 1 (momentum), defined on the manifold. I want to integrate the tensor on a local neighborhood around a point p.

More precisely, I want to know if the linear momentum is conserved on that part of the manifold using only intrinsic quantities. In a book, they say that they define a parallel vector field on that neighborhood so they can do a dot product between the two tensors. But I'm really not convinced of this method because there no "absolute" parallel field on surface and each field will give different result.

How can I do that?

 

[zhentil]

You can't integrate 1-forms on a 2-manifold. If you want to find out whether it's conserved, you can either integrate it around every possible closed path, or find out whether it's closed and use Stokes's theorem.

 

[siyphsc]

wouldnt your local neighborhood be locally euclidean?

 

[GPPaille]

siyphsc:
Yes but I consider a finite part of the surface, where curvature can't be ignored.

zhentil:
I know that I can't integrate a 1-form, that's why they define another vector field so they do a dot product and obtain a scalar. Integrating this scalar is supposed to give the momentum component in the direction of the vector field. The ultimate goal is to fully understand the derivation of the Navier-Stokes equations on a manifold, and the only book I know do it the way I explained above. They use Stokes theorem, here's what they say:

$$T^{ab}$$ is the stress tensor, $$V^{a} (resp. A^a)$$ is the velocity (resp. acceleration) of the fluid and $$\rho$$ is the density. Let an arbitrary parallel field of covariant surface vectors $$l_b$$ be defined on the surface. If we make a balance of linear momentum in the direction of the parallel field,

$$\frac{d}{dt}\int_S{\int{\rho V^a l_a dA}} = \int_S{\int{\rho A^a l_a dA}}=\oint_C{T^{ab}m_a l_b ds}$$

Where S is a finite part of the surface, C is the boundary of S and $$m_a$$ is a vector normal to the curve C and tangent to S. Using Stokes theorem we have

$$\int_S{\int{\left[\rho A^a - T^{ab}_{,b}\right] l_a dA}} = 0$$

Since S and $$l_a$$ are arbitrary,

$$\rho A^a = T^{ab}_{,b}$$

So the first question, is it valid (and why). If it's not, how can I obtain the same kind of result using a correct derivation. Integrating over all closed loops seems hard to define mathematically.