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.

This is a companion discussion topic for the original entry at

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?

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 :slight_smile:

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:

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?

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?


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

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.

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.

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”.


you gotta love that guy’s hat in that video

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?

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

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

killer explanation!

@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!

@afshar: Hi, the unit for circulation corresponds to F® dr. If F® is the force, then


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


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.

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).