What will be the reduced form of the Stokes' theorem for multiply connected regions (with n holes)?
If S is a surface in space with n holes with boundaries of the holes as $C_1, C_2, \dots, C_n$ then
$$\int\int_S \nabla \times F\cdot n\, dS=\sum_{i=1}^n \int_{C_i} F.dr$$
The outer circle $C$ is positively oriented, and the inner circles $C_i$ negatively
Is this true?
- Please correct if not true.
- Also give a good reference where one can study Stokes' theorem for such surfaces.
An intuitive explanation of why it works.
You can decompose S into n+1 surfaces with no hole: the boundary between surface i and surface i+1 touches hole i twice (I hope this is understandable without a drawing).
Then the left hand-side double integral is the sum of n+1 same integrals over each of the n+1 sufaces; and the right hand-side is the sum of n+1 same integrals over the n+1 boundaries, because at each frontier between two surfaces the two integrals cancel each other.
So the usual Stokes formula can be applied separately to each of the n+1 surfaces, and the sum of these n+1 equations gives the result.