[…] Vector Calculus: Gradient, Flux, Divergence, Curl & Circulation […]
Nice breakdown into simple terms. There were two statements that seemed contradictory thouugh.
1.Conservative fields have zero curl.
2…consider a river: it’s field is conservative.
Therefore a river must have zero curl. If that is true, based on the two aforementioned premises.
My question is, how can you place a paddle wheel in a river and have it turn, since is has zero curl?
Thanks for your answer.
Rocky
@Rocky: Thanks for the feedback, I should put in a diagram to show what I mean.
As Scotty mentions above, the paddlewheel is meant to be in the 2d (horizontal) plane. So rather than a paddle wheel like you have on the riverboats, imagine the axis pointing straight up, and the paddles sticking out of the water like a wall. The top and bottom half get pushed equally so it doesn’t turn.
Kalid,
The reason we have energy at all is not a big mystery. The sum of energy in the universe is still zero, conservation of energy was never violated. If you recall from physics, some energies are negative, such as gravitational potential energy. So we have 1, and -1, but it’s still zero as a whole.
But why {1,-1} instead of 0? Because it’s more disordered and therefore more stable. But then why order? Because the universe is expanding faster than entropy can fill it so it is forced to become more cold, stable, and orderly on local scales.
I’d suggest taking a look Victor Strenger’s stuff, it’s amazing and goes into a lot more detail.
I am quite confused between circulation & rotation. Does the existence of circulation means a rotational flow?
Shouldn’t circulation = Work?
Force x distance = work
Force.dr = work
@Jayson: Yep, circulation is often interpreted as work in the physics sense (force x distance). However, circulation is a more general concept which can apply to movement in any vector field (i.e, it doesn’t have to be “force” which is multiplied by distance).
Hey, I still don’t understand curl and I was looking for some help with it. I’ve seen this visualization before in texts and such and it doesn’t work for me, particularly when trying to do E&M with it.
I think that the explanation of circulation, which is pretty straightforward, detracts a little from the explanation of curl. Firstly, the whirlpool example is little confusing, because with a boat in water, as soon as you match the water’s speed, the force is zero. So it’s not analogous to circulation in curling vector fields (i.e. the magnetic field) and confuses the definition of conservative. In fact, a boat twisting in a whirlpool IS conservative. The water does work on the boat which is stored in rotational kinetic energy which could then be extracted from the boat. It’s not analogous to energy gain due to a curling field because in such a field the force is perpendicular to the field and not in the direction of the path.
A much better example is when you first talked about a boat moving AROUND the whirlpool, i.e. anti-centripetal force that only changes direction and whose work can never be re-extracted from the system because the force is always perpendicular to the path of motion (inward), just like a magnetic field creating a cyclotron. I can’t say that I have a good enough intuition to know for sure, but the river going in different speeds right next to each other seemed to make sense too. In any case, when I then try to “shrink down” the concept to an infinitely small area to get an intuition of curl, I can’t wrap my head around the role of area in circulation.
I understand path integrals, and I get calculus, but I still don’t understand the physical meaning of circulation per unit area. What area is it? The area inside the path? Since area drops off with the square of the size, how can you shrink it the same as you can path length?
What I can’t see is how the dotting of a vector along an infinitely small path (i.e. the circulation) in a continuous field can not produce a zero result in an infinitely small path where the field, assuming a smooth field gradient, becomes essentially constant. The whole reason that circulation in a field need not be conservative is because, like you said, the field does not need to be symmetric, and can follow the path of circulation to an extent. But it seems like shrinking down to an infinitely small path would void this possibility.
Finally, how do you relate the concept of curl to the concepts of divergence and/or cross-product as in the latter del-cross-F notation? Is this in another discussion? The relationship is not making sense to me without the math.
Thanks so much and sorry for such a long post!
Hello Kalid. I cannot wrap my head around line integrals that are in velocity fields. When the path of the line integral travels through a force field we can interpret F.dr = dw, where dw = (small amount of work done by the force in the direction of travel)
However how can we interpret F.dr if F is a velocity field?
(Velocity).dr doesn’t make any intuitive sense to me.
Is there a way to visualize this?
thanks for any help
[…] excerpt: A conservative field is “fair” in the sense that work needed to move from point A to […]
hi kalid
i just wanted to know that since the curl is given by circular force/area…will the surface area of an object dipped in the flowing water vary the curl experienced by it. That is will a smaller object with a smaller area experience greater curl than a larger object?
This question came to me because i somehow feel that curl is analogus to pressure. Am i going wrong somewhere?
Binnoy
A month has gone by, with no response. No thoughts from anyone on this?
It’s always fun reading your thoughts, Kalid – I can quit trying to decipher the dry hieroglyphics of formal math and give my imagination and intuition free rein.
I’ve been thinking about the curl product. Picture a 3D orthonormal set of axes, set in a scalar field. Now introduce a force, say in the x direction – now there’s a vector field. Do the same in the y direction – now there’s curl. But does it matter whether the “primary force” and the “sideways” force are x and y respectively, or y and x instead? Are there two different curls, dependant on your point of view?
And then, introduce a third force from underneath, the z direction, which could give rise to six different curls.
This is getting hairy. A couple of more dimensions, and I’d have a full head.
Any barbers out there?
For line integrals, I don’t recall the formal proof, but here’s my intuition. We’re trying to get from A to B along some path (imagine B is “uphill” from A). If different paths took different amounts of energy, then we could go from A to B along cheap path, and “roll downhill” on the expensive one. This would be a loop back to A with a net gain in energy.
This is possible in theory, but not for conservative fields (and most fields in the real world are conservative). For example, gravity is conservative, so any work you do to lift a rock is returned when you let it roll downhill, or let it drop instantly. Either way, energy in = energy out (ignoring friction). Put another way, I guess you could define a conservative field to be ones where this relation holds (every path from A to B takes the same net energy).
A non-conservative field is something like a whirlpool. We could put a rudder on an axis, vertically in the pool. The endpoint of the rudder would turn one revolution (returning to its starting point), but we got free energy out of it (turning the axis). From an intuitive/physics perspective, there’s “something” which is spending energy to run the whirlpool, and we’re just capturing some of it.
There’s a bit of a chicken-and-egg problem: the path integrals are independent because the field is conservative, and the field is conservative because the path integrals are independent :). That said, there are quick checks (curl = 0) to see whether a field truly is conservative [curl = 0 is another way of saying there’s no net rotation in the field, i.e., there are no “mini-paths” anywhere, even at the infinitesimal level].
Can you show how to derive the identity curl X curl(F)= del(del.F) - delsquared F?
I hope you can forgive my terminology as I do not know how to get the upside down triangle symbol on my PC.
I LOVE THIS! I’m 13 and this website makes me understand it all! ^^
[…] Grad, Flux and Curl (if you already know vector […]
Hi Gordon, thanks for the suggestion – I’ll have to dive into that some day :).
Can you do an article giving some insight as to why the line integral of a gradient field is path independent? The proof is simple and straightforward but my lack of intuition behind me has bugged me for quite some time.