Comments on: Vector Calculus: Understanding Circulation and Curl http://betterexplained.com/articles/vector-calculus-understanding-circulation-and-curl/ Learn Right, Not Rote. Fri, 10 Feb 2012 02:50:41 +0000 hourly 1 http://wordpress.org/?v=3.2.1 By: Binnoy http://betterexplained.com/articles/vector-calculus-understanding-circulation-and-curl/#comment-6775 Binnoy Fri, 28 Oct 2011 17:42:13 +0000 http://betterexplained.com/articles/vector-calculus-understanding-circulation-and-curl/#comment-6775 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 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

]]> By: Like, part 2 « Arcsecond http://betterexplained.com/articles/vector-calculus-understanding-circulation-and-curl/#comment-1562 Like, part 2 « Arcsecond Sat, 30 Oct 2010 09:53:16 +0000 http://betterexplained.com/articles/vector-calculus-understanding-circulation-and-curl/#comment-1562 [...] excerpt: A conservative field is “fair” in the sense that work needed to move from point A to [...] [...] excerpt: A conservative field is “fair” in the sense that work needed to move from point A to [...]

]]> By: Jayson http://betterexplained.com/articles/vector-calculus-understanding-circulation-and-curl/#comment-1561 Jayson Wed, 28 Jul 2010 19:20:20 +0000 http://betterexplained.com/articles/vector-calculus-understanding-circulation-and-curl/#comment-1561 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 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

]]> By: Mike http://betterexplained.com/articles/vector-calculus-understanding-circulation-and-curl/#comment-1560 Mike Wed, 23 Jun 2010 15:42:34 +0000 http://betterexplained.com/articles/vector-calculus-understanding-circulation-and-curl/#comment-1560 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! 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!

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