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Mathematical Expression Editor

Divergence measures the rate field vectors are expanding at a point.

While the gradient and curl are the fundamental “derivatives” in two dimensions,
there is another useful measurement we can make. It is called divergence. It measures
the rate field vectors are “expanding” at a given point.

The divergence of a vector field

Let’s state the definition:

Given a vector field , where the divergence is given by Some authors use the
notation for the divergence of a vector field.

Is the divergence of a vector field a scalar or a vector?

vector.scalar.neither
a vector nor a scalar.

The divergence is a number that tells you how much the field is expanding at a point.
However, no directional information is given.

Consider the vector field . Compute:

Consider the vector field . Compute:

What does divergence measure?

As we’ve already said, divergence measures the rate field vectors are expanding at a
point. To be more explicit, the divergence measures how the magnitude of
the field vectors change as you move in the direction of the field vectors:

And:

The most obvious example of a vector field with nonzero divergence is :

On the other hand, recall that a radial vector field is a field of the form
where where is a real number. The divergence of these vector fields can be
surprising.

Compute the divergence of .

First note
and next note
so

Compute the divergence of .

Now we will see a radial vector field with zero divergence.

Consider the vector field :

Compute .

Since this is a radial field, we know its divergence is when .

Now let’s see a radial vector field with negative divergence.

Consider the vector field :

Compute .

Since this is a radial field, we know its divergence is when .

Measuring flow across a curve

Let be a vector field, be a smooth vector valued function tracing a curve exactly
once as runs from to ,

Recall that the line integral measures the accumulated flow of a vector field along a
curve. We see this because measures how “aligned” field vectors are with the
direction of the path . On the other hand, if we set then for any given value of , is
a vector that is orthogonal to . Moreover, given a closed curve, where is
parameterized with the interior on the left, points outward. Below we see a
curve along with some tangent vectors and some outward normal vectors :

Since measures how “aligned” field vectors are with
vectors orthogonal to the direction of the path, the integral
measures the flow of a vector field across a curve. Some folks
call this a flux integral. Since and , we may write as

Consider the following vector field and curve parameterized by :

Do you expect to be positive, zero, or negative?

positivezeronegative

Consider the following vector field and curve parameterized by :

Do you expect to be positive, zero, or negative?

positivezeronegative

Consider the following vector field and curve parameterized by :

Do you expect to be positive, zero, or negative?

positivezeronegative

With our next example, we’ll get our hands dirty.

Below we see a closed curve along with some representative field vectors of a vector
field :

Setting or (depending on the direction of the curve), estimate:

If , we
have that: We know that , and . Now we’ll compute for each edge of the
triangle above. For the bottom edge and . So we have: Along the right
edge, and . So Finally, along the left edge, and . So Hence we estimate

Let and let be the unit circle centered at the origin. Compute

The path can be
parameterized by
with . To compute the integral, write with me

Connections to Green’s Theorem

Finally, note that if , then:

We also see that
this leads us to the flux form of Green’s Theorem:

Green’s Theorem If the components of have continuous partial derivatives and is a
boundary of a closed region and parameterizes in a counterclockwise direction with
the interior on the left, and , then

Let be a vector field with . Compute:

Suppose that the divergence of a vector field is constant, .