Surface Integral: Right Side = Left Side?

[pardesi]

May be this should have been in math section but since this came out while studying Electrodynamics i put it here
we have
$$\boxed{\int_{S} \nabla \times \vec{B}.d\vec{a}=\oint \vec{B}.d\vec{l}}$$

Q.well there are many areas with the same boundary which one to choose from?

well if we know the area the boundary is fixed but not vice-versa does only the right side equal left but nor always the left side equals right.
Can someone explain

 

[CompuChip]

It doesn't matter which surface you take, that's the beauty of it. For example, if the curve on the right hand side is a circle, depending on the symmetry it may be easiest to take either a flat disk, or a half-sphere, or anything else.

There's probably a nice proof for it too, but I wouldn't be able to give you that by heart.

 

[pardesi]

wow
tat's gr8

 

[captphoton]

It doesn't matter which surface you take, that's the beauty of it. For example, if the curve on the right hand side is a circle, depending on the symmetry it may be easiest to take either a flat disk, or a half-sphere, or anything else.

There's probably a nice proof for it too, but I wouldn't be able to give you that by heart.

Stokes's Theorem.