Curl and Circulation

Circulation is the amount that a force pushes along a closed boundary; it can be seen as the twisting or turning that a force applies. A vector field can be the source of this circulation.  If you had a paddlewheel lying in your boundary, the circulation would be the amount that it got turned.  The more circulation, the more twisting force you have.  

Curl is simply the circulation per unit area.  If you have a lot of twisting force in a little area then you are going to have a large curl.  If you have a low circulation on a large area, then you will have a small curl.  Of course, zero circulation means zero curl.

Intuition and More Details

Curl is the amount of twisting or turning force in a vector field. 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.  We put a small paddle wheel inside the water and notice if it turns. If it does, this particular field has a curl.  If it doesn't turn, there is no curl.

What does it mean if the paddle turns?  It means the water is pushing harder on one side of the paddle than the other, causing it to twist.  The larger the curl the more forceful the twist.  A turning paddle wheel also indicates that the field is uneven and asymmetrical; if it were even then it would push on all sides equally and the paddle wouldn't turn. 

The field is not conservative. A conservative field is "fair" in the sense that the amount of work needed to move from A to B, along any path, is equal. In a field with curl, you can get a "free ride" by moving in the direction of the curl, while you have to fight the field if you move the opposite direction. Thus, conservative fields therefore have zero curl because they are fair.

The most familiar example of a conservative field is gravity. Technically, it takes the same amount of work to lift a rock from point A to B; the change in potential energy is the same.

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.  The direction is a little more tricky: it is the direction you want to point the axis of your paddlewheel in order to get the maximum rotation. In other words, it is the direction which will give you the most "free work" from the field.

But wait, aren't there two directions?  Couldn't you just turn the paddle 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.

So, we have our definition of curl.  Let's look at some funky examples and see if they have any curl or not.

Mathematics

The circulation is the integral of a vector field along a path. In essence, you are adding up how much the field pushes in the direction of the path.  What does this mean?  We should expect a dot product, because we want to know the amount that one vector is pushing in the direction of another. The two vectors we need are (1) the vector in the direction of the path and (2) the direction the field is pushing.

If we have a function for position, we can take its time derivative and get the velocity at any point (1).  The velocity vector is always in the direction of the movement, so therefore it is in the direction of the path. You should understand why we aren't going to use the position vector itself: it tells us where we are, but not the direction we are going. 

The force vector (2) is specified by the field we are in. No derivatives or any other manipulations are necessary.
Our formula for circulation is:

Remember that dr is the derivative of the position vector r, which is the velocity vector we want.

Recall that curl is the circulation per unit area (taking the limit as Area goes to 0).  However, the curl is a vector, and its direction is the normal to the surface with the vector field.  Its magnitude is as before:  circulation/area. Remember how we assigned the direction of the vector: counterclockwise circulation gives a curl pointing out of the page. Thus, we have an expression for curl:

If I have been successful there should be no confusion about how we got either of these formulae (fancy Latin plural). They spring up naturally from our definition of circulation and curl. Mathematics should be a tool for succinctly expressing what we already know.

Examples

I'm going to put down some vector fields, lets examine their circulation and curls. Woo-hoo!