Green’s Theorem is a fundamental theorem of calculus.

Function | Example | Derivative | Interpretation |

explicit curve | slope of the tangent line | ||

vector-valued function | tangent vector | ||

explicit surface | the vector that points in the initial direction of greatest increase of | ||

implicit curve (level curve) | gradient vectors are orthogonal to level sets | ||

In this section we will learn the *fundamental derivative* for two-dimensional vector
fields, as well as a new fundamental theorem of calculus.

### The curl of a vector field

Calculus has taught us that knowing the derivative of a function can tell us
important information about the function. In a similar way we have that seen that if
we wish to understand a function of several variables , then the gradient, , contains
similar useful information. If you have a vector field we now ask: “what is the
natural analogue of a derivative in this setting?” When the vector field is two or
three-dimensional, the *curl* is the analogue of the derivative that we are looking
for:

**curl**is given by In three dimensions, given a vector field , where the

**curl**is given by

Other authors sometimes use the notation for the scalar curl of a two-dimensional vector field , and for the curl of a three-dimensional vector field .

Now for something you’ve seen before, but in a different form.

#### What does the curl measure?

The curl of a vector field measures the rate that the direction of field vectors “twist” as and change. Imagine the vectors in a vector field as representing the current of a river. A positive curl at a point tells you that a “beach-ball” floating at the point would be rotating in a counterclockwise direction. A negative curl at a point tells you that a “beach-ball” floating at the point would be rotating in a clockwise direction. Zero curl means that the “beach-ball” would not be rotating. Below we see our “beach-ball” with two field vectors. If

In essence, the scalar curl measures how the magnitude of the field vectors change as you move to the right, in a direction perpendicular to the direction of the field vectors:

In our next example, we see a field that has local rotation (nonzero curl) but does not have global rotation.

**every**point. We’ll assume that the magnitudes of the vectors are constant along vertical lines. Set . To estimate , we examine the change in between and : and we should also check the change of between and : Averaging these values together we find To estimate , we examine the change in between and : and we should also check the change of between and : Averaging these values together we find So we approximate So field above example does not have global rotation, but it does have local rotation.

Now we’ll show you a field that has global rotation but no local rotation!

Finally, we’ll show you a field that has global rotation with local rotation in the
*opposite* direction!

#### Curl in three dimensions

In this section we introduce curl in three dimensions. You already know how to compute it:

What does it mean? Well first of all, in three dimensions, curl is a *vector*. It points
along the axis of rotation for a vector field. You should think of a tornado:

At this point we only know how to take the derivative (via the curl) of a vector field of two or three dimensions. You can take another course to learn more about derivatives of -dimensional vector fields.

### A new fundamental theorem of calculus

Recall that a fundamental theorem of calculus says something like:

To compute a certain sort of integral over a region, we may do a computation on the boundary of the region that involves one fewer integrations.

In the single variable case we have:

*Green’s Theorem*:

The upshot is that we were able to use Green’s Theorem to transform a tedious integral into a trivial one.

How is Green’s Theorem a fundamental theorem of calculus? Well consider this:

#### Strategy for evaluating line integrals

At this point we have three ways to evaluating two-dimensional line integrals:

- Direct computation.
- The Fundamental Theorem of Line Integrals.
- Green’s Theorem.

How do you know which method to use? Here are some rules of thumb.

Identify the field With line integrals, we must have a vector field. You must identify this vector field.

Compute the scalar curl of the field If the scalar curl is zero, then the field is a gradient field. If the scalar curl is “simple” then proceed on, any you might want to use Green’s Theorem.

Is the boundary a closed curve? If the field is a gradient field and the curve is closed, then the integral is zero (by both the Fundamental Theorem of Line Integrals and Green’s Theorem).

If the boundary is not a closed curve and the curl is not zero, you must use direct computation.

If the curl is simple and the boundary is a closed curve, then maybe use Green’s Theorem.

#### The shape of things to come

Recalling that is the set of differentiable functions from to where **all** of the
derivatives are continuous, we can make the following “chain” of derivatives:

Moreover, by the Clairaut gradient test, the scalar curl of the gradient vector is zero. Given a function , we can follow through the chain to see what happens: