Is Stokes' Theorem Intuitively Satisfying and Accurate?

[Nikitin]

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]

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 :)

The "Grad" operator is the (formal) vector of partial derivatives. The cross product tells you how to apply the vector part to the vector valued function F. Compare that with the Divergence. It isn't necessarily intuitively pleasing to everyone. Your intuition could tell you 2+2=5. Your intuition is trained by your experience. You have to do the math to get a "good" intuition about what should happen. Intuition is not knowledge from the aether, it is a "quick calculation" from your past experience. When your intuition leads you counter to facts you need to train it by generating experience. If your intuition tells you 2+2=5, you need to do some arithmetic counting on your fingers until you see that's wrong and your intuition will adjust and tell you 2+2=4. If your intuition tells you Stokes theorem isn't right, you need to work some problems and see how the devil within the details conspires to keep Stokes theorem valid.

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.

 

[Nikitin]

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.

 

[jambaugh]

I think it also helps to look at the physical application of fluid flow understanding that the curl of the velocity field gives you vorticity. In that context the "flux of vorticity across the surface equals the flow around the perimeter". But that's really providing one with a better intuition about the meaning of the Curl rather than about the truth of Stokes' thm.