Vector Calculus: Understanding Circulation and Curl

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Circulation is the amount of force that pushes along a closed boundary or path. It's the total "push" you get when going along a path, such as a circle.

A vector field is usually the source of the circulation. If you had a paper boat in a whirlpool, the circulation would be the amount of force that pushed it along as it went in a circle. The more circulation, the more pushing force you have.

Curl is simply the circulation per unit area, circulation density, or rate of rotation (amount of twisting at a single point). Imagine shrinking your whirlpool down smaller and smaller while keeping the force the same: you'll have a lot of power in a small area, so will have a large curl. If you widen the whirlpool while keeping the force the same as before, then you'll have a smaller curl. And of course, zero circulation means zero curl.

Intuition

Circulation is the amount of "pushing" force along a path. Curl is the amount of pushing, twisting, or turning force when you shrink the path down to a single point. Let's use water as an example.

Suppose we have a flow of water and we want to determine if it has curl or not: is there any twisting or pushing force? To test this, we put a paddle wheel into the water and notice if it turns (the paddle is vertical, sticking out of the water like a revolving door -- not like a paddlewheel boat):

If the paddle does turn, it means this field has curl at that point. If it doesn't turn, then there's no curl.

What does it really mean if the paddle turns? Well, it means the water is pushing harder on one side than the other, making it twist. The larger the difference, the more forceful the twist and the bigger the curl. Also, a turning paddle wheel indicates that the field is "uneven" and not symmetric; if the field were even, then it would push on all sides equally and the paddle wouldn't turn at all.

The fact that there is a "twist" means the field is not conservative (this has nothing to do with its political views).

A conservative field is "fair" in the sense that work needed to move from point A to point B, along any path, is the same. For example, consider a river: its field is conservative. Sure, you can get a free ride downstream, but then you have to do work to get back to your starting point. Or, you can do work to move upstream, and get a free ride back. Either way, the amount of work you "put in" is the same as what you get back.

However, in a field with curl (like a whirlpool), you can get a free ride by moving in the direction of the twist. In a whirlpool, you can get a free trip by moving with the current in a circle. If you fight the current and go the wrong way, you have to use energy with no free ride at all.

Conservative fields have zero curl: there are no free twists to push you along. Alternatively, if a field has curl, it is not conservative.

Gravity is another example of a conservative field. Technically, if you lift a rock and then let it fall, the energy you get from falling is the same as what you put in to lift the rock. Theoretically speaking, no energy was gained or lost in this transaction.

Additional Details

To be technical, curl is a vector, which means it has a both a magnitude and a direction. The magnitude is simply the amount of twisting force at a point.

The direction is a little more tricky: it's the orientation of the axis of your paddlewheel in order to get maximum rotation. In other words, it is the direction which will give you the most "free work" from the field. Imagine putting your paddlewheel sideways in the whirlpool - it wouldn't turn at all. If you put it in the proper direction, it begins turning.

But wait a minute -- aren't there two directions to get a twisting motion? Couldn't you just turn the paddlewheel "upside down" and get the maximum curl as well?

Yep, you're right. By convention alone, if the paddle wheel is rotating counterclockwise, its curl vector points out of the page. This is a type of right-hand rule: make a fist with your right hand and stick out your thumb. If the circulation/pushing force follows the twisting of your fingers (counterclockwise), then the curl vector will be in the direction of your thumb.

Mathematics

Circulation is the integral of a vector field along a path - you are adding how much the field "pushes" you along a path.

How do we find this? Well, we should expect some type of dot product, because we want to know the amount that one vector (the force) is pushing in the direction of another (the path). So, the two vectors we need are (1) the path vector and (2) the field vector at every point along the path.

If we have a function that defines the position at any time, ( F(t) ), we can take the time derivative to get the velocity at that position.

The velocity vector is always in the direction of movement -- if you are moving from A to B, the velocity vector will be an arrow from A to B, i.e. your change in position or your direction of movement. So, we can use the velocity to get our direction.

It's important to understand why we aren't using the position vector itself -- it tells us where we are, but not where we're going. We need to know our direction to see how much "push" we are getting: Knowing your position in a river isn't important -- are you going upstream or downstream, and at what angle?

The force vector (2) is defined by the field we are in. No derivatives or other changes are necessary -- every point in the field has some force acting on it.

So, our formula for circulation is:

\displaystyle{\text{ Force at position r } = F(r)}

\displaystyle{\text{ Direction at position r } = dr }

\displaystyle{\text{ Total pushing force = Circulation } = \int F(r) \cdot dr }

Remember, velocity is simply the derivative of position (r), so (dr) is a vector giving us our direction. We integrate along the entire path and use the dot product to see how much pushing force is applied. We then sum up these "pushes" to get the total circulation.

Since curl is the circulation per unit area, we can take the circulation for a small area (letting the area shrink to 0). However, since curl is a vector, we need to give it a direction -- the direction is normal (perpendicular) to the surface with the vector field. The magnitude is the same as before: circulation/area.

Recall that by convention (a bunch of people agreeing), counterclockwise circulation will give a curl pointing out of the page. Using these facts, we can create the formula for curl:

\displaystyle{ Curl = \frac{circulation}{area} = \frac{\int F(r) \cdot dr}{\int S} }

Where (S) is the surface we are considering; the direction of the curl is the normal to the surface.

You'll see fancier equations for curl where the surface shrinks to zero (such as in wikipedia), but recognize the basic intuition -- curl is the circulation per unit area.

Parting Thoughts

You'll often see curl of a field (F) written like this:

\displaystyle{ Curl(F) = \nabla \times F }

which is a cross-product of the gradient and the field (F). This has to do with how curl is actually computed, which will be material for another article (and probably in your textbook already -- see wikipedia for details).

If I have been successful, you should understand intuitively what circulation and curl mean, and how we got the formulae above. They spring up naturally from our definition of circulation as "pushing force along a path" and curl as "pushing force/area".

Math should be a tool for clearly stating what we already know. Understand the intuition and then tackle the complicated formulas. Happy math.

PS. Have some fun and check out this video of a famous whirlpool. Imagine the circulation on this (go on, imagine):

Other Posts In This Series

  1. Vector Calculus: Understanding Flux
  2. Vector Calculus: Understanding Divergence
  3. Vector Calculus: Understanding Circulation and Curl
  4. Vector Calculus: Understanding the Gradient
  5. Understanding Pythagorean Distance and the Gradient
  6. Vector Calculus: Understanding the Dot Product
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42 Comments

  1. I’m terrible at math, I got a D in Calculus in high school, but I like the conceptual visualization of mathematical principles.

    You wrote:
    “However, in a field with curl (like a whirlpool), you can get a free ride by moving in the direction of the twist. In a whirlpool, you can get a free trip by moving with the current in a circle. If you fight the current and go the wrong way, you have to use energy with no free ride at all.”

    How is it possible to get a free ride at all if energy can neither be created nor destroyed?

    For example, if i tossed a paper boat into a body of water with a whirl pool in the middle, the boat could get a “free ride” around the whirl pool without using any of its own power, but wouldn’t there be a catch (literally) in that in order for the boat to get this free ride, it would be trapped in the vortex of the whirlpool itself, doomed to eventually be sucked into the center?

    Would then the energy required to escape the pull of the whirlpool offset the energy gained by the “free ride” from one side of the whirlpool to the other?

    I know scientists can get these free rides somehow because they use the gravitational pull of the sun to slingshot spacecraft like voyager 1 into the outer solar system. Does that mean that energy can be created from gravity without investing an equal amount of energy to begin with? Is there a way we can Harvest gravity?

  2. Hi Tom, great question. I had wondered the whirlpool as well — it seems like we can get a free ride, right?

    However, the whirpool needs to be created by something, such as a propeller or water going down a hole. For a propeller, energy is being used to create the whirlpool, and the boat is just feeding off that. In the case of water draining down a sink, water is flowing downwards (losing energy), and the boat is feeding off that change.

    Other examples of “free” energy sources are solar power and wind: we can get something for nothing. However, the sun is the ultimate power source for those events, and we just tap into the energy the sun gives off.

    Gravity is interesting — we can get “free” energy depending on how things are positioned initially. For example, if I go to a mountain, I can make “free” energy by rolling the rocks that are there downhill. Something did the work to get rocks up there (continental drift pushing land to make the mountain), and I get to use it for free.

    In a similar way, our planet started off far away from other ones. We can “slingshot” by falling towards the other planet, which accelerates us more and more. As far as I know, the slingshot doesn’t actually change the speed, but can change the direction of motion (so you get a free change in direction). And the reason we can do the slingshot at all is because we initially started away from the planet.

    I guess the summary is that we can’t create “free” energy, but we can use energy that is already in the system, like the sun or starting off far away from other planets. It’s a bigger, hairier question to ask why we have energy in the universe at all, instead of it all being a blank void.

    To answer your question: I’d say we can harvest gravity, in the sense of extracting energy from objects that have already been separated (like rocks on top of a mountain). However, this extracts energy that is already in the system (still useful), rather than creating new energy. Eventually we’d run out of rocks to topple over :)

  3. Thanks for your answer. That makes a lot of sense. Your analogy of running out of rocks to topple over has me thinking though. The energy in that scenario is derived from the distance between the rock and the ground. But if that rock is sitting on the top of a lever, it still falls the same distance to reach the ground, but multiplies its force at the other end of the lever. Could that multiplied force then be stored as energy and used again to reset the rock back at the top of the hill? After all, the force was multiplied due to leverage before it was stored, yet the distance down is the same as the distance back up.

    Lets say you have a heavy rock and a light metal spring both spring suspended at an equal height above the ground. Since any two objects fall to the earth at the same speed regardless of mass, they cover the same distance in the same amount of time. Yet the heavier object has more potential energy before the fall than the lighter object.

    Now the spring has the ability to store energy. If the weight of the rock is placed on a lever it would multiply the force that could be applied to the spring. Then if then both objects were brought together at the same point at on the ground by gravity (such as the rock rolling down the lever to the spring)Would it then have more stored energy to reset the rock back at the same height?

    If you used one rock and two springs and ramp/levers then the weight of the spring could reset one lever while the rock rolled down the other.

    Here is an example of what I mean:

    http://photos1.blogger.com/blogger/1436/1318/1600/MACHINE.jpg

    When the ball gets to the bottom, the spring shoots it to the top of the other ramp. The leverage then sets the spring as the level falls a little. meanwhile the weight of the spring on the other ramp resets that lever back into position. After the spring is set the ball releases and rolls down to repeat the process back the other way.

    Would this keep the world in supply of rocks to topple?

  4. that was a fab. explanation:can anyone pl. tell me what if there is any conclusion that can be derived from curl and divergence values..say if a field has zero curl and non-zero divergence..(or any other combo :both 0 , both not 0 etc.) can we infer something?

  5. So what’s the units of measure when you get an answer to the “Curl”. For instance in your whirlpool example, or in magnetism?

  6. Hi Tom, great question. I believe the units depend on what field you are considering.

    Circulation: Path integral
    Curl: Circulation per unit area

    In fluid dynamics, circulation has units “length squared over time” because you are taking a path integral (length) across a velocity (length/time).

    The corresponding curl would then have units (length^2/time) / (length^2). That’s probably the meaning in the whirlpool example :).

    In electricity & magnetism, the units would be different. Taking a look at Maxwell’s equations, the curl of in E (electric field in volts/meter) is dB/dt (change in magnetic field over time, or tesla/second).

    Great question, I hadn’t thought about this that much before, always expecting the units to “work out”. The units depend on what field you are measuring to find circulation.

  7. AFAIK, slingshot (or “gravity assist”, to give the technical name) can in fact give the spacecraft energy – it comes from the motion of the planet. So the planet is slowed down while the spacecraft speeds up. Because the planet is much more massive, however, its change of speed is much smaller than the change of speed of the tiny spacecraft.

  8. what is the physical meaning of the circulation when the vector field is velocity of water? you mentioned that the circulation has units “length squared over time” , what this unit is for? as the case when vector field is “force” the circulation means “work”.

  9. Thanks a lot.. I am a million times clearer about what a curl is now…
    But i still have a doubt- why should the curl of conservative field be zero. Cant a conservative field have unsymmetrical distribution of vectors which can lead them to have curl?

  10. Thanks a lot.. now my concept of curl are a million times better..
    But still I couold not uunderstand why exactly the curl of a non conoservative field cannot be zero.. I mean, conservative vector field need not be symmetrical. mAnd any vector field that is unsymmetrical can actually cause the circulation?
    please clearify this point.
    thanks a lot
    Swetha

  11. thanx 4 ur brilliant simplification of vector calculus. makes studyn em fields and waves way more intrstn when u actually have a virtual image of what the equations represent. chris from kenya

  12. @Swetha: For a field to be conservative, it means there is “no free ride” and therefore every loop you take has no “overall push”. If this is the case, it means curl (circulation / area) is zero at every point. A non-conservative field could have zero curl at some point, and non-zero curl at the others (it’s the non-zero areas which make it non-conservative).

    I see it as a conservative field means “no free ride”. So, if there is a free ride anywhere in the field, it is not conservative. Hope this helps!

    @Christian: Glad you enjoyed it! I completely agree, having a mental visualization of what’s happening makes math so much easier.

    @Tuguldur: Thanks!

  13. @afshar: Hi, the unit for circulation corresponds to F(r) dr. If F(r) is the force, then

    F(r)dr

    would have units (force * distance) which is indeed work.

  14. What a great explanation. I can tell you my class has learned all about curl and divergence but very few of my classmates right now can conceptually understand what a curl really is. My book explained the whole curl concept in a paragraph and my teacher also gave a brief explanation because he never fully understood it in the first place.

  15. Absolutely fantastic. With this, the concepts are finally coming together.
    One little thing…
    Explain that the ‘paddlewheel’ as a horizontal disk in the flow. I was fighting try to picture is as the wheel on the back of one of those boats.

    I went to the Wiki site with the images, and it helped trigger the realization that the paddle is supposed to be horizontal (XY plane).
    http://en.wikipedia.org/wiki/Curl_(mathematics)

  16. 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

  17. @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.

  18. 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.

  19. @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).

  20. 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!

  21. 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

  22. 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

  23. 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?

  24. 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.

  25. 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.

  26. 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].

  27. I am in Calculus III right now, and I found this post to be helpful in order to better understand curl intuitively. I really find the math in this class fascinating, as it seems to be the most practical of any math class I’ve taken so far. It’s really neat how we can model real life forces that on the surface seem so complex with relative ease! Thanks again for this post.

  28. I find it disconcerting that pseudovectors–aka rotational vectors (such as curl)–are continually referred to as vectors, blurring the distinctions between the two types of construction and causing much confusion and heartache for the uninitiated. A true vector’s direction is determined straightforwardly; a rotational vector’s direction is a matter of convention. The two have different symmetries, different multiplicative properties, and so on. Especially on a site like yours that I depend on for clarity in visual thinking, I would love to see a clear distinction drawn between the two. I feel that, as time goes on, things like this lumping of really very distinct categories of things need to gradually fall by the wayside in order to foster deeper, more direct, and more intuitive understanding.

  29. Chris, I have some sympathy for your view that the concept of vector used here is too general. But this is a site which tries (and succeeds!) to make numerous poorly understood concepts simple. To do that, the intricacies found when any mathematical idea is pursued at length have to be trimmed, so concepts such as co/contra variant vectors, bi-vectors, pseudovectors, and their extensions in topics such as tensors and n-dimensional geometries have to be ignored. For example, in a different field, atoms don’t really(?) look like minature solar systems, but the concept has been useful in teaching thousands of chemistry students.

    Getting the foundations down, even if they’re not perfectly accurate, is the most important thing. Of course, cautions can be made that the model used is a model only, it’s useful only up to a point. Korzybski’s idea that the map is not the territory is apt – (remember AE Van Vogt’s “World of Null-A”?).

  30. (the paddle is vertical, sticking out of the water like a revolving door — not like a paddlewheel boat).

    @khalid
    Can you link me to an image so that i can understand it more clear.
    Thanks

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