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: 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):

• Points in the direction of greatest increase of a function
• Is zero at a local maximum or local minimum (because there is no single direction of increase)

The term gradient (grad) typically refers to the derivative of vector functions, or functions of more than one variable. Yes, you can say a line has a gradient (its slope), but using the term gradient for single-variable functions is unnecessarily confusing. Keep it simple.

“Gradient” can refer to gradual changes of color, but we’ll stick to the math definition if that’s ok with you. You’ll see the meanings are related.

Properties of the Gradient

Now that we know the gradient is the derivative of a multi-variable function, let’s derive some properties.

The regular, plain-old derivative gives us the rate of change of a single variable, usually x. For example, dF/dx tells us how much the function F changes for a change in x. But if a function takes multiple variables, such as x and y, it will have multiple derivatives: the value of the function will change when we “wiggle” x (dF/dx) and when we wiggle y (dF/dy).

We can represent these multiple rates of change in a vector, with one component for each derivative. Thus, a function that takes 3 variables will have a gradient with 3 components:

has one variable and a single derivative:

has three variables and three derivatives:

The gradient of a multi-variable function has a component for each direction.

And just like the regular derivative, the gradient points in the direction of greatest increase (trust me on this, I’ll create a derivation a bit later ). However, now that we have multiple directions to consider (x, y and z), the direction of greatest increase is no longer simply “forward” or “backward” along the x-axis, like it is with functions of a single variable.

If we have two variables, then our 2-component gradient can specify any direction on a plane. Likewise, with 3 variables, the gradient can specify and direction in 3D space to move to increase our function.

A Twisted Example

I’m a big fan of examples to help solidify an explanation. Suppose we have a magical oven, with coordinates written on it and a special display screen:

We can type any 3 coordinates (like “3,5,2″) and the display shows us the gradient of the temperature at that point.

The microwave also comes with a convenient clock. Unfortunately, the clock comes at a price — the temperature inside the microwave varies drastically from location to location. But this was well worth it: we really wanted that clock.

With me so far? We type in any coordinate, and the microwave spits out the gradient at that location.

Be careful not to confuse the coordinates and the gradient. The coordinates are the current location, measured on the x-y-z axis. The gradient is a direction to move from our current location, such as move up, down, left or right.

Now suppose we are in need of psychiatric help and put the Pillsbury Dough Boy inside the oven because we think he would taste good. He’s made of cookie dough, right? We place him in a random location inside the oven, and our goal is to cook him as fast as possible. The gradient can help!

The gradient at any location points in the direction of greatest increase of a function. In this case, our function measures temperature. So, the gradient tells us which direction to move the doughboy to get him to a location with a higher temperature, to cook him even faster. Remember that the gradient does not give us the coordinates of where to go; it gives us the direction to move to increase our temperature.

Thus, we would start at a random point like (3,5,2) and check the gradient. In this case, the gradient there is (3,4,5). Now, we wouldn’t actually move an entire 3 units to the right, 4 units back, and 5 units up. The gradient is just a direction, so we’d follow this trajectory for a tiny bit, and then check the gradient again.

We get to a new point, pretty close to our original, which has its own gradient. This new gradient is the new best direction to follow. We’d keep repeating this process: move a bit in the gradient direction, check the gradient, and move a bit in the new gradient direction. Every time we nudged along and follow the gradient, we’d get to a warmer and warmer location.

Eventually, we’d get to the hottest part of the oven and that’s where we’d stay, about to enjoy our fresh cookies.

Don’t eat that cookie!

But before you eat those cookies, let’s make some observations about the gradient. That’s more fun, right?

First, when we reach the hottest point in the oven, what is the gradient there?

Zero. Nada. Zilch. Why? Well, once you are at the maximum location, there is no direction of greatest increase. Any direction you follow will lead to a decrease in temperature. It’s like being at the top of a mountain: any direction you move is downhill. A zero gradient tells you to stay put – you are at the max of the function, and can’t do better.

But what if there are two nearby maximums, like two mountains next to each other? You could be at the top of one mountain, but have a bigger peak next to you. In order to get to the highest point, you have to go downhill first.

Ah, now we are venturing into the not-so-pretty underbelly of the gradient. Finding the maximum in regular (single variable) functions means we find all the places where the derivative is zero: there is no direction of greatest increase. If you recall, the regular derivative will point to local minimums and maximums, and the absolute max/min must be tested from these candidate locations.

The same principle applies to the gradient, a generalization of the derivative. You must find multiple locations where the gradient is zero — you’ll have to test these points to see which one is the global maximum. Again, the top of each hill has a zero gradient — you need to compare the height at each to see which one is higher. Now that we have cleared that up, go enjoy your cookie.

Mathematics

We know the definition of the gradient: a derivative for each variable of a function. The gradient symbol is usually an upside-down delta, and called “del” (this makes a bit of sense – delta indicates change in one variable, and the gradient is the change in for all variables). Taking our group of 3 derivatives above

Notice how the x-component of the gradient is the partial derivative with respect to x (similar for y and z). For a one variable function, there is no y-component at all, so the gradient reduces to the derivative.

Also, notice how the gradient can itself be a function!

If we want to find the direction to move to increase our function the fastest, we plug in our current coordinates (such as 3,4,5) into the equation and get:

So, this new vector (1, 8, 75) would be the direction we’d move in to increase the value of our function. In this case, our x-component doesn’t add much to the value of the function: the partial derivative is always 1.

Obvious applications of the gradient are finding the max/min of multivariable functions. Another less obvious but related application is finding the maximum of a constrained function: a function whose x and y values have to lie in a certain domain, i.e. find the maximum of all points constrained to lie along a circle. Solving this calls for my boy Lagrange, but all in due time, all in due time: enjoy the gradient for now.

The key insight is to recognize the gradient as the generalization of the derivative. The gradient points to the maximum of the function; follow the gradient, and you will reach the local maximum.

Divergence (div) is “flux density”—the amount of flux entering or leaving a point. Think of it as the rate of flux expansion (positive divergence) or flux contraction (negative divergence). If you measure flux in bananas (and c’mon, who doesn’t?), a positive divergence means your location is a source of bananas. You’ve hit the Donkey Kong jackpot.

Remember that by convention, flux is positive when it leaves a closed surface. Imagine you were your normal self, and could talk to points inside a vector field, asking what they saw:

• If the point saw flux entering, he’d scream that everything was closing in on him. This is a negative divergence, and the point is capturing flux, like water going down a sink.
• If the point saw flux leaving, he’d sniff his armpits and say all flux was existing. This is a positive divergence, and the point is a source of flux, like a hose.

So, divergence is just the net flux per unit volume, or “flux density”, just like regular density is mass per unit volume (of course, we don’t know about “negative” density).

The bigger the flux density (positive or negative), the stronger the flux source or sink. A div of zero means there’s no net flux change in side the region. In plain english:

Divergence = Flux / Volume

Math Intuition

Now that we have an intuitive explanation, how do we turn that sucker into an equation? The usual calculus way: take a tiny unit of volume and measure the flux going through it. We need to add up the total flux passing through the x, y and z dimensions.

Imagine a cube at the point we want to measure, with sides of length dx, dy and dz. To get the net flux, we see how much the X component of flux changes in the X direction, add that to the Y component’s change in the Y direction, and the Z component’s change in the Z direction. If there are no changes, then we’ll get 0 + 0 + 0, which means no net flux.

If there is some change in the field, we get something like 1 -2 +5 (flux increases in X and Z direction, decreases in Y) which gives us the divergence at that point.

In pseudo-math:

Total flux change = (field change in X direction) + (field change in Y direction) + (field change in Z direction)

Or in more formal math:

Assuming F1 is the field in the X direction, F2 in the Y and F3 in the Z.

A few remarks:

• The symbol for divergence is called “del” and is an upside down triangle.
• Divergence is a single number, like density.
• Divergence and flux are closely related – if a volume encloses a positive divergence (a source of flux), it will have positive flux.
• “Diverge” means to move away from, which may help you remember that divergence is the rate of flux expansion (positive div) or contraction (negative div).

Divergence isn’t too bad once you get an intuitive understanding of flux. It’s really useful in understanding in theorems like Gauss’ Law.