Why stokes theorem is applied to only open surfaces?

Why stokes theorem is applied to only open surfaces?


1 Answers

askIITians Faculty 74 Points
8 years ago
Hello student,
Please find the answer to your question below
Stokes theorem is always applied to open surfaces becausea closed surface is a surface that has no boundary.
In other words, a closed surface S has no “edge” floating around. Another way to say this isthat its boundary is empty: ∂S = ∅. In general, the boundary of a surface will be a curve, orpossibly several curves.
A consequence of Stokes’ theorem is that integrating a vector field which is a curl along aclosed surface S automatically yields zero.
In case the idea of integrating over an empty set feels uncomfortable – though it
shouldn’t – here is another way of thinking about the statement. If S is a closed surface, cut itinto two parts S1 and S2 along some curve C. For example, we can cut the sphere S into theupper hemisphere S1 and lower hemisphere S2 along the equator C. Applying Stokes’s theoremto each part yields= 0
where the opposite signs come from the orientation convention.
In fact, property characterizes curls: A vector field is the curl of some vector field if andonly if its integral along any closed surface is zero.
So stokes theorem is applied only to open surfaces.

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