**Introduction**

Two key concepts in vector calculus are **divergence**
and **curl, **the latter of which is
sometimes called *circulation. * Basically, **divergence** has to do with how a vector field changes its magnitude in
the neighborhood of a point, and **curl **has
to do with how its direction changes. Look at the graph of **Field 1 **below. It has both **zero divergence** and **zero
curl**, because it doesn’t change in either magnitude or direction in
the neighborhood of any point. For
instance, if you look near the origin, you’ll see that all the little
arrows have the same direction and the same length. (Actually, since this is a **constant** field, it will be true near
any point you pick, but in general fields can have different properties at
different points.)

**Field
1**

**Divergence**

Now look at **Field 2**:

**Field
2**

At the origin, all arrows point radially outward. Go out about half a unit along any
radius, and you’ll see that the arrows are still pointing in the same
direction, but they’re getting longer. Since the length of an arrow in the plot
is proportional to the strength of the field in the neighborhood of the arrow,
we can see that as we go farther away from the origin along any radius, the
direction of the field isn’t changing, but its strength is. The graph in **Field 2 **above is actually a plot of the vector field **F(x,y) =
{x, y}.**. For this vector field, the divergence of **F**, written **div F(x,y), **is equal to **2.** Symbolically, **div F = 2. **

What if you’re not at the origin?
Well, then, you just pick some other point and look at the **net change** in the field at that
point. Pick (1,1) for
instance. What you’ll see
there is that the arrows heading **toward**
the point are shorter than those heading **away**
from the point, so the change in the strength of the field is positive—in
other words, the divergence of the field is positive. You’ll see the same result at any
point—it’s just easier to see at the origin where there are no
arrows heading in, so that all you have to do is look at the outward-pointing
ones.

Now, although **Field 2** is changing
in both the *x* and *y *directions, you can have one like **Field 3** below, where the arrows are
only changing in length in one direction/component and not the other:

**Field
3**

In this case, the y-component of the length of
each arrow is always 1, but the x-component gets larger in magnitude as we get
farther away from the origin. And
in fact, the field that produced this plot is Its divergence
is **1.**

We can, of course, also have fields with **negative divergence:**

**Field
4**

The basic difference here is that the arrows are
all pointing **toward** the origin
instead of away from it. The divergence of the field in **Field 4** is **-2. **Or, if you pick some point other
than the origin, notice that the lengths of the arrows heading into that point
are longer than the lengths of the arrows heading away from the point. Pick (1,1), for instance, to make sure
you see this.

Of course, the divergence of a field doesn’t
have to be constant, either. The
divergence of **Field 5** for instance
is *2x + 2y: *

**Field
5**

And here’s one last field whose divergence
you should consider:

**Field
6**

For **Field
6 **above:

1.
Decide whether the divergence is positive,
negative, or zero in each of the four quadrants. Explain your reasoning.

**Curl**

All the vector fields you’ve seen so far
have had **zero curl**. It’s time
to look at some that don’t.

**Field
7**

If you look at the origin, where the field seems
to “circulating” around the point, you can see why **curl** used to be called **circulation.** Notice that at any distance from the
origin, say at 1 unit, the arrows have constant length but constantly changing
direction. Since their length is
constant, the divergence of **Field 7**
is zero; since the direction is always changing, the **curl** is non-zero. To put it another way, the vectors
all ** curl **around a central point. (Clever nomenclature, eh?) The
vector field of which

A field with **negative curl**, btw, will simply curl in
the opposite (clockwise) direction, like **Field
8** below:

**Field
8**

You might think from these examples that all vector fields with a non-zero curl show this “curling around a point” behavior. That would be a reasonable conclusion given the examples you’ve seen, but it’s actually incorrect. Here’s an example where it looks like the vectors all fall on diagonal lines, with no perceptible curving, yet there is a non-zero curl:

**Field
9**

The way to think about **curl** then
is a little more general than just vectors curling around a point. Still, some idea of
“circulation” must be involved since that’s what this
phenomenon used to be called. A
good way to think about curl, it turns out, is to think of a vector field as
representing a fluid that actually flows and then to imagine a little
paddlewheel placed at a point in the flow.
If the paddlewheel will turn, then the field has a non-zero curl at that
point. In the case of **Field 9**, put a little paddlewheel at
the origin. Then the arrows on the
left will push up on the paddles on the left and the arrows on the right will
push down on the paddles on the right, which will produce a net imbalance of
forces and hence a **torque**, and thus
the paddle will spin. Hence, the
field has a nonzero curl at the origin.

You don’t even have to have direction changes anywhere for a field to have nonzero curl, though. Let’s look at a field where the arrows are everywhere pointing in the same direction:

**Field
10**

There is still a non-zero curl in **Field 10—**in fact, a positive
one. You can see that by imagining
your paddlewheel at (1,1). The
arrows on the right-hand side of the wheel are longer than those on the
left-hand side, though, so there will be more force pushing the wheels on the
right. This imbalance of forces
leads to a torque and thus to the wheel’s spinning counterclockwise and
hence to a positive curl. In fact, **Field 10** is the plot of and **curl G = 1.**

Now, technically, since **curl** represents a direction, it has to be represented by a vector,
and it’s more correct to say that . If you’re
conversant with torque, you know that it acts in a direction perpendicular to
the plane of the forces causing the spin, so it wouldn’t surprise you
that **curl G** is in the *z-*direction for **Field 10.**

1.
Although **curl** is a vector quantity, **div**
is not—it’s a scalar.
Go back to what you learned about **divergence**
in the first part of this lab to explain why in fact you should *expect* it to be a scalar.

**Now, let’s move on to two more complicated examples:**

Here is a vector field with both divergence *and* curl positive:

**Field
11**

**Field 11 **is a plot of the vector
field **{x - y, x + y}.** Notice
that it has both "counterclockwise curling" and "outward
streaming" and hence both positive curl and positive divergence.

Finally, look at **Field 12** below,
whose description is **{x – y ^{2 }+ 2, xy + y}**:

**Field
12**

Things here are quite complicated, but to simplify just a bit, look in the immediate neighborhood of the red point. What you see there are positive divergence and negative curl. Every one of the nearby vectors points outward, so there is no question about the divergence’s being positive. As for the curl, notice that all of the closest vectors are rotated clockwise from pointing straight out, so there is no question about the curl’s being negative either..

What makes this particular field more complex than the others is that both
the divergence and the curl depend on the values of *x *and *y *at the point of
interest. So, though the curl near
the red point is negative, the curl in the neighborhood of the blue point is
positive. Without doing any calculations,
you can see that the curl of this vector field depends on the sign of its *y-*coordinate
at any point.

Now, **divergence**
and **curl** are measures of *how a vector field changes*, so we know
they’re going to involve derivatives. And that observation takes us to the
next lab.