Can the Curl be Calculated over 2-Manifolds?

[Jhenrique]

If I can compute the divergence over an closed curve (or an area) and too over an closed surface (ie, an volume) so I can, actually, to compute the divergence over a n-manifold.
$$\nabla \cdot \vec{F} = \lim_{\Delta V → 0} \frac{1}{\Delta V} \oint_{S} \vec{F} \cdot \hat{n} dS$$$$\nabla \cdot \vec{F} = \lim_{\Delta A → 0} \frac{1}{\Delta A} \oint_{C} \vec{F} \cdot \hat{n} dr$$So, make sense speak about the curl of a 2-manifold? Do you undestand my ask? Exist a math formlation for the curl over a curve
$$(\nabla \times \vec{F}) \cdot \hat{n}= \lim_{\Delta S → 0} \frac{1}{\Delta S} \oint_{C} \vec{F} \cdot d\vec{r}$$but, exist for a surface?

 

[jvicens]

As a fellow scientist, I can certainly understand your question. The concept of computing the divergence over a n-manifold is a valid one, as long as the manifold in question is smooth and well-defined. This concept is often used in the study of differential geometry and vector calculus.

To answer your question about the curl over a 2-manifold, the formula you have presented is indeed correct. The curl of a vector field over a 2-manifold is defined as the limit of the circulation of the vector field around a small closed curve on the surface, divided by the area enclosed by the curve. This can be generalized to higher dimensions as well.

In terms of a mathematical formulation for the curl over a surface, there are several ways to approach it. One way is to use the surface integral of the vector field, which can be expressed as a double integral over the surface. Another way is to use the cross product of the tangent vectors of the surface to compute the curl. Both approaches are valid and can be used depending on the specific problem at hand.

In summary, both the divergence and curl can be computed over n-manifolds, as long as the manifold is well-defined and smooth. The formulas you have presented are correct and can be applied to various problems in physics and mathematics. I hope this helps clarify your question.