Stokes' Theorem says the curve integral of any surface S simply equals the counter-clockwise circulation around its boundary-curve C.

How can this be right? Let's say you have a hemisphere surface S with centre in origo, and its shadow on the xy plane. Both surfaces will have C as their boundary curve, and so according to the theorem they will have equal circulation. However, what if the curl increases with z? Then more curl will go through the higher surfaces than the bottom, and thus the curl-integral of surface S will not equal the circulation around C.

In addition, can somebody explain to me why the formula for curl, ∇xF, is intuitively pleasing? Ie, why does it make sense?

Thanks :)

jambaugh
Gold Member
Stokes' Theorem says the curve integral of any surface S simply equals the counter-clockwise circulation around its boundary-curve C.
Careful there, it does not say that. It says that the surface integral of the curl of a vector field equals the line integral of that field around the boundary. It is a vector version of the fundamental theorem of calculus. Integration and differentiation can cancel leaving only the evaluation on the boundary.

How can this be right? Let's say you have a hemisphere surface S with centre in origo, and its shadow on the xy plane. Both surfaces will have C as their boundary curve, and so according to the theorem they will have equal circulation. However, what if the curl increases with z? Then more curl will go through the higher surfaces than the bottom, and thus the curl-integral of surface S will not equal the circulation around C.
Remember that the two surfaces must have the same boundary and that both the surfaces and the curls have relevant orientation information. Finally remember that not just any vector field is the curl of another vector field. See e.g. Helmholtz decomposition. Every vector field has a solenoidal and conservative component. Your arbitrarily increasing the field w.r.t. z doesn't mean that component is from the curl of another field. Try to construct an example and see what prevents your generating a counter example to Stokes' theorem.

In addition, can somebody explain to me why the formula for curl, ∇xF, is intuitively pleasing? Ie, why does it make sense?

Thanks :)

It doesn't hurt to also plod through the details of the proof of Stokes' thm. Have you looked at Green's theorem which is the planar form of Stokes'? Do you understand that you can chop a piecewise smooth continuous surface up into tiny pieces which look almost planar? Have you seen how the boundaries of all the tiny pieces cancel each other leaving only integration around the whole boundary?

What is left is to consider how the vector curl dots with the vector surface normal to see how varying the surface doesn't change this relationship.

Careful there, it does not say that. It says that the surface integral of the curl of a vector field equals the line integral of that field around the boundary. It is a vector version of the fundamental theorem of calculus. Integration and differentiation can cancel leaving only the evaluation on the boundary.
Apologies, I meant curl integral, not curve integral. The latter makes no sense anyway :p

And yeh, I understand Stokes' theorem now. I just needed to look at the surface as a bunch of infinitesimal rectangles, where each has a curl, and where all but the edging rectangles get their curl negated.

But the formula for curl I still struggle with.. I've looked over the proof, and I understand it, but it isn't intuitive at all, unlike the formula for divergence. I dunno, I guess if you say it's not intuitive, I'll have to accept that and just memorize the formula.

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jambaugh