Vector Calculus: Understanding Circulation and Curl
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. If it 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: it’s 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:
Force at position r = 
Direction at position r = 
Total pushing force = 
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:
Curl = 
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:
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):
Tags: area, circulation, curl, field, intuition, Math, surface, vector calculus
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Have a question? Know an explanation that caused your own a-ha moment? Write about it here.
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?
tom — February 20, 2007 @ 4:12 pm
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
Kalid — February 20, 2007 @ 6:28 pm
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?
tom — February 21, 2007 @ 2:13 am
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?
kirtika — September 25, 2007 @ 11:27 pm
THANKS MUCH BEEN LOOKING FOR THIS FOR YEARS
WILL — February 7, 2008 @ 5:40 pm
So what’s the units of measure when you get an answer to the “Curl”. For instance in your whirlpool example, or in magnetism?
Tom Nygaard — February 27, 2008 @ 5:42 am
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.
Kalid — February 27, 2008 @ 9:43 pm
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.
sabik — May 3, 2008 @ 3:02 am
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”.
afshar — July 6, 2008 @ 3:06 am
“I SURVIVED LAKE PEIGNEUR”
you gotta love that guy’s hat in that video
Anonymous — July 20, 2008 @ 11:12 am