Creative Analogies for Curl, Div, and Flux with Examples from Calc IV

[daytripper]

Can someone give me analogies for each of these? I know the standard ones so try to be creative. I just received an A- in Calc IV and these words are KILLING me (moreso Curl and Div than Flux, as I'm close to understanding those and I've no idea what flux is). It would help if you gave examples in the context of Stokes' theorem and the Divergence theorem. I have a fairly good grasp on the ideas of the surface integrals of scalar/vector fields (though I prefer vector).
btw, by Stokes' theorem, I suppose I mean Kelvin-Stokes theorem (the curl theorem). Stokes' theorem, as it's stated on wikipedia (I know, I know... it's all I have) makes sense as the generalization of the familiar fundamental theorem of calculus.
p.s. In doing research in order to properly post this, I think I understand flux as the amount of whatever to move through a surface per unit time (ambiguous). This definition seems to restrict my understanding to 3-space and I'd like a more general one... if possible.
Thank you

edit: and, I realize this is a lot and some of you are probably wondering how I passed at this point, but I also am not sure of the explicit meaning of "conservative vector field". I know a bit about what that implies, but not exactly what it means. thanks again.

 

[NoMoreExams]

Have you tried wikipedia? They usually try to come up with an explanation.

 

[daytripper]

NoMoreExams,

I just read "Curl" and "Divergence" articles on wikipedia, but these concepts aren't exactly what I have trouble with. Curl was a terrific article and quickly strengthened my understanding. Divergence, not so much (I understand the density example but a question about incompressibility arose upon reading this which I'll detail a little later).

As I said, my problem isn't necessarily the ideas of Curl and Divergence, but more the theorems that they're used in (namely the "Kelvin-Stokes' Theorem" and the "Divergence Theorem"). I've tried to explain the Kelvin-Stokes' Theorem to myself using the following analogy, so let me know if this is correct or not.
Consider a region M living in 2-space. Liken the surface inside this region to that of a table-top. Now imagine putting a layer of water on the table that magically doesn't fall off unless pushed and can pull more water into it from the edges, equally magically =]. As I understand the Curl Theorem (Kelvin-Stokes'), it says that were you to cause a current in this water, the total amount of water to spill over the edge of the table (∂M) will be equal to the sum of... rotations (if that makes sense) in the water. Is this right or is this analogy off the mark?

Moving on to Divergence, wikipedia says that divergence can be a sum of sources and sinks in a vector field. It also says a fluid is incompressible if at every point the divergence is zero and in that case, there can be "no net flow... across any closed surface". What about the vector field <1,0,0>? div(<1,0,0>) = 0 but there's a net flow across the surface y=0, as the fluid is running in the positive x direction. Or is it saying strictly "across" (tangent to) a surface and I'm saying "through"?

I had real the Curl and Divergence theorem articles on wikipedia to the best of my ability but didn't think to try searching for simply "Curl" --disambiguation--> math. The curl article really strengthened my understanding and the whole idea of sticking a "paddle wheel" into the graph makes sense. I guess the three dimensional analogue would be a paddle... ball (trapped in a cage rather than rotating on hinges)? Using that analogue, do you think the curl of a 3-dimensional vector field would be tangential to the ball? Or would that vector be in 4-space and pointless to try to visualize (I'm pretty sure it's not)?
edit: I've decided that this vector would live in 3-space and be essentially the axis on which the "paddle ball" rotates. Is this correct? Is there significance to the magnitude of this vector? (surface integrals of vector fields confused the hell out of me until I realized what the magnitude of (a x b) meant)

Thank you. Sorry for the exceedingly long post.
-Tim

 

[daytripper]

anyone? my understanding is good enough for course work but not myself

 

[George Jones]

anyone? my understanding is good enough for course work but not myself

Try the book div, grad, curl and all that by H. M. Schey.

 

[daytripper]

thanks, george, I'll look into that

 

[swraman]

I watched all this professors webcasts even though I was in a different prof's class. He is pretty good at explaining, but I don't know if he goes into the kind of conceptual detail you are looking for.

http://webcast.berkeley.edu/course_details_new.php?seriesid=2008-D-54472&semesterid=2008-D

Lecture 36, at about 9 minutes it starts and talks about Greens Thm and Curl, conservative fields, and diergence, gives somewhat visual definition.

You can look at some of the other lectures too, I believe Flux is lecture 40 at the beginning.

As for how Stokes and Div Thms work...I just think of them as "tools" to solve integrals. Cant offer any help there.

Hope this helps

 

[mathwonk]

take a surface in space and a vector with foot on that surface. then the vector makes a certain angle with that surface, which can be measured by taking the dot product of the vector with a unit vector normal to the surface. recall the dot product equals thee cosine of that angle times the product of the lengths of the two vectors.

this dot product measures the length of that part of the original vector pointing right out of the surface, or the tendency of the vector to flow across the surface, which may be called flux or flow.

If you have a vector field, i.e. a family of vectors, one at each point of the surface, you can perform this dot product operation at every point and integrate the result, getting an average tendency of that field to flow across the surface.

those famous theorems say that if the vector field is also defined on as well as inside of a closed surface, then there is something else that can be integrated over the interior of the surface that will give the same answer as the integral measuring flow across the surface.

intuitively, these theorems say something like, if you have a incompressible fluid, as we may assume is true of water, then the total flow across the surface must be causEd somehow by a source of water originating inside the surface.

 

[daytripper]

As for how Stokes and Div Thms work...I just think of them as "tools" to solve integrals. Cant offer any help there.

Hope this helps

Yea, I've gotten that far. I know how to use them and such but I really would enjoy a deeper understand of what's going on there. I'll look at those videos. I hope they help. Thanks for your help.

 

[daytripper]

intuitively, these theorems say something like, if you have a incompressible fluid, as we may assume is true of water, then the total flow across the surface must be causEd somehow by a source of water originating inside the surface.

Thank you for the definition on flux but I wish you would have read one of my posts. I think I said to ignore my request for information on flux, because I understood it. All well, at least it increased my confidence in my understanding, so thank you anyway.

as far as the second half of your post, now we're getting somewhere. haha. That's the divergence theorem, right? that makes complete sense. thank you. If the fluid is incompressible and if there's a source inside a closed region of space, then the total divergence of that closed region will be equal to the flux across the surface which encloses it. correct? am I correct in saying this is the divergence theorem?

can you offer similar insight for the curl theorem? (aka kelvin-stokes' theorem)

thanks again.

 

[mathwonk]

why don't you use the same ideas i just gave yourself (i.e. dot product) to explain curl?

 

[daytripper]

Ok, I'll give it a shot.
The curl theorem says integral of the curl of a vector field across a surface is equal to the line integral of a vector field on the boundary of that surface. Would it be true to say that the only rotational tendency that matters is on the boundary of the surface? See, there's something fundamental missing. With divergence, it was the idea of an incompressible fluid needing to diverge from inside a surface in order to cause a flux (due to displacement) but with this, I don't see it. My analogue of curl is sticking a paddle wheel into a surface and seeing how it spins. I don't see how this effect translates to an effect on the boundary.
-Tim

 

[mathwonk]

well if you stir a barrel full of water. won't the water at the edges of the barrel move around?

the point is that is you measure the circulation of a vector field around a CURVE, by dotting with the tangent vectors to the curve and integarting.

then the theorem says that th circulation around a clsoed curve bounding a sufrgace,

can also be measured by ointegrating something else over the surface, and then that something else is called the curl of the vector field.

what that is of course is a local tendency to rotate. it is detected by focusing on a small disc centered at a point and integrating around the boundary of the disc. if you get a positive number for smaller and smaller discs, then there must be some rotation occurring at that point.

again some kind of rigidity seems to be in effect for an effect to be propagated to neighboring points, analogous to incompressibility.

in effect, you can detect the presence of whirlpools inside a surface bounded by a closed curve, from noticing the tendency of points on the curve to move around the curve.

 

[arildno]

Think about "curl" as the circulation about a tiny area enclosing a point.

Then, the theorem just says that if you add all contributions to the net curl over some area, then that yields the net circulation around the edge of the area.

 

[Defennder]

So the reason why the theorem makes sense is because any rotation in middle of the surface gets canceled out by a neighbouring rotation, so the only ones that matter are the ones at the surface boundary which doesn't get canceled out?

 

[dx]

Read the first chapters of Feynman Lectures Vol. II for a great introduction these ideas.

 

[arildno]

So the reason why the theorem makes sense is because any rotation in middle of the surface gets canceled out by a neighbouring rotation, so the only ones that matter are the ones at the surface boundary which doesn't get canceled out?

exactly

 

[daytripper]

So the reason why the theorem makes sense is because any rotation in middle of the surface gets canceled out by a neighbouring rotation, so the only ones that matter are the ones at the surface boundary which doesn't get canceled out?

he got to it before I could. thanks everyone for the help. I understand these three ideas now... "it's a good thing"