APEX Flow, Flux, Green’s Theorem and the Divergence Theorem

Section 15.4 Flow, Flux, Green’s Theorem and the Divergence Theorem

Subsection 15.4.1 Flow and Flux

Line integrals over vector fields have the natural interpretation of computing work when

\vec{F}

represents a force field. It is also common to use vector fields to represent velocities. In these cases, the line integral

\int_{C} \vec{F} \cdot d \vec{r}

is said to represent flow.

Link to full-sized image

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Figure 15.4.1. Illustrating the principles of flow and flux

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Let the vector field

\vec{F} = ⟨ 1, 0 ⟩

represent the velocity of water as it moves across a smooth surface, depicted in Figure 15.4.1. A line integral over

C

will compute “how much water is moving along the path

C .

”

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In the figure, “all” of the water above

C_{1}

is moving along that curve, whereas “none” of the water above

C_{2}

is moving along that curve (the curve and the flow of water are at right angles to each other). Because

C_{3}

has nonzero horizontal and vertical components, “some” of the water above that curve is moving along the curve.

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When

C

is a closed curve, we call flow circulation, represented by

\oint_{C} \vec{F} \cdot d \vec{r} .

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The “opposite” of flow is flux, a measure of “how much water is moving across the path

C .

” If a curve represents a filter in flowing water, flux measures how much water will pass through the filter. Considering again Figure 15.4.1, we see that a screen along

C_{1}

will not filter any water as no water passes across that curve. Because of the nature of this field,

C_{2}

and

C_{3}

each filter the same amount of water per second.

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The terms “flow” and “flux” are used apart from velocity fields, too. Flow is measured by

\int_{C} \vec{F} \cdot d \vec{r},

which is the same as

\int_{C} \vec{F} \cdot \vec{T} d s

by Definition 15.3.2. That is, flow is a summation of the amount of

\vec{F}

that is tangent to the curve

C .

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By contrast, flux is a summation of the amount of

\vec{F}

that is orthogonal to the direction of travel. To capture this orthogonal amount of

\vec{F},

we use

\int_{C} \vec{F} \cdot \vec{n} d s

to measure flux, where

\vec{n}

is a unit vector orthogonal to the curve

C .

(Later, we’ll measure flux across surfaces, too. For example, in physics it is useful to measure the amount of a magnetic field that passes through a surface.)

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Figure 15.4.2. Further terminology for curves: closed, simple, positively oriented

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How is

\vec{n}

determined? We’ll later see that if

C

is a closed curve, we’ll want

\vec{n}

to point to the outside of the curve (measuring how much is “going out”). We’ll also adopt the convention that closed curves should be traversed counterclockwise.

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(If

C

is a complicated closed curve, it can be difficult to determine what “counterclockwise” means. Consider Figure 15.4.3. Seeing the curve as a whole, we know which way “counterclockwise” is. If we zoom in on point

A,

one might incorrectly choose to traverse the path in the wrong direction. So we offer this definition: a closed curve is being traversed counterclockwise if the outside is to the right of the path and the inside is to the left.)

Link to full-sized image

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Figure 15.4.3. Determining “counterclockwise” is not always simple without a good definition

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When a curve

C

is traversed counterclockwise by

\vec{r} (t) = ⟨ f (t), g (t) ⟩,

we rotate

\vec{T}

clockwise 90

^{\circ}

to obtain

\vec{n} :

\vec{T} = \frac{⟨ f^{'} (t), g^{'} (t) ⟩}{‖ \vec{r}^{'} (t) ‖} \Rightarrow \vec{n} = \frac{⟨ g^{'} (t), - f^{'} (t) ⟩}{‖ \vec{r}^{'} (t) ‖} .

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Letting

\vec{F} = ⟨ M, N ⟩,

we calculate flux as:

\begin{aligned} \int_{C} \vec{F} \cdot \vec{n} d s & = \int_{C} \vec{F} \cdot \frac{⟨ g^{'} (t), - f^{'} (t) ⟩}{‖ \vec{r}^{'} (t) ‖} ‖ \vec{r}^{'} (t) ‖ d t \\ = \int_{C} ⟨ M, N ⟩ \cdot ⟨ g^{'} (t), - f^{'} (t) ⟩ d t \\ = \int_{C} (M g^{'} (t) - N f^{'} (t)) d t \\ = \int_{C} M g^{'} (t) d t - \int_{C} N f^{'} (t) d t . \end{aligned}

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As the

x

and

y

components of

\vec{r} (t)

are

f (t)

and

g (t)

respectively, the differentials of

x

and

y

are

d x = f^{'} (t) d t

and

d y = g^{'} (t) d t .

We can then write the above integrals as:

\begin{aligned} = \int_{C} M d y - \int_{C} N d x . \end{aligned}

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This is often written as one integral (not incorrectly, though somewhat confusingly, as this one integral has two “

d

’s”):

\begin{aligned} = \int_{C} M d y - N d x . \end{aligned}

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We summarize the above in the following definition.

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Definition 15.4.4. Flow, Flux.

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Let

\vec{F} = ⟨ M, N ⟩

be a vector field with continuous components defined on a smooth curve

C,

parametrized by

\vec{r} (t) = ⟨ f (t), g (t) ⟩,

let

\vec{T}

be the unit tangent vector of

\vec{r} (t),

and let

\vec{n}

be the clockwise 90

^{\circ}

degree rotation of

\vec{T} .

The flow of $\vec{F}$ along $C$ is

$\int_{C} \vec{F} \cdot \vec{T} d s = \int_{C} \vec{F} \cdot d \vec{r} .$
The flux of $\vec{F}$ across $C$ is

$\int_{C} \vec{F} \cdot \vec{n} d s = \int_{C} M d y - N d x = \int_{C} (M g^{'} (t) - N f^{'} (t)) d t .$

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This definition of flow also holds for curves in space, though it does not make sense to measure “flux across a curve” in space.

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Measuring flow is essentially the same as finding work performed by a force as done in the previous examples. Therefore we practice finding only flux in the following example.

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Example 15.4.5. Finding flux across curves in the plane.

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Curves

C_{1}

and

C_{2}

each start at

(1, 0)

and end at

(0, 1),

where

C_{1}

follows the line

y = 1 - x

and

C_{2}

follows the unit circle, as shown in Figure 15.4.6. Find the flux across both curves for the vector fields

{\vec{F}}_{1} = ⟨ y, - x + 1 ⟩

and

{\vec{F}}_{2} = ⟨ - x, 2 y - x ⟩ .

Link to full-sized image

(a)

Link to full-sized image

(b)

Figure 15.4.6. Illustrating the curves and vector fields in Example 15.4.5. In (a) the vector field is

{\vec{F}}_{1},

and in (b) the vector field is

{\vec{F}}_{2} .

Solution.

We begin by finding parametrizations of

C_{1}

and

C_{2} .

As done in Example 15.3.12, parametrize

C_{1}

by creating the line that starts at

(1, 0)

and moves in the

⟨ - 1, 1 ⟩

direction:

{\vec{r}}_{1} (t) = ⟨ 1, 0 ⟩ + t ⟨ - 1, 1 ⟩ = ⟨ 1 - t, t ⟩,

for

0 \leq t \leq 1 .

We parametrize

C_{2}

with the familiar

{\vec{r}}_{2} (t) = ⟨ \cos t, \sin t ⟩

0 \leq t \leq π / 2 .

For reference later, we give each function and its derivative below:

{\vec{r}}_{1} (t) = ⟨ 1 - t, t ⟩, \vec{r}_{1}^{'} (t) = ⟨ - 1, 1 ⟩ .

{\vec{r}}_{2} (t) = ⟨ \cos t, \sin t ⟩, \vec{r}_{2}^{'} (t) = ⟨ - \sin t, \cos t ⟩ .

When

\vec{F} = {\vec{F}}_{1} = ⟨ y, - x + 1 ⟩

(as shown in Figure 15.4.6.(a)), over

C_{1}

we have

M = y = t

and

N = - x + 1 = - (1 - t) + 1 = t .

Using Definition 15.4.4, we compute the flux:

\begin{aligned} \int_{C_{1}} \vec{F} \cdot \vec{n} d s & = \int_{C_{1}} (M g^{'} (t) - N f^{'} (t)) d t \\ = \int_{0}^{1} (t (1) - t (- 1)) d t \\ = \int_{0}^{1} 2 t d t \\ = 1 . \end{aligned}

Over

C_{2},

we have

M = y = \sin t

and

N = - x + 1 = 1 - \cos t .

Thus the flux across

C_{2}

is:

\begin{aligned} \int_{C_{1}} \vec{F} \cdot \vec{n} d s & = \int_{C_{1}} (M g^{'} (t) - N f^{'} (t)) d t \\ = \int_{0}^{π / 2} ((\sin t) (\cos t) - (1 - \cos t) (- \sin t)) d t \\ = \int_{0}^{π / 2} \sin t d t \\ = 1 . \end{aligned}

Notice how the flux was the same across both curves. This won’t hold true when we change the vector field.

When

\vec{F} = {\vec{F}}_{2} = ⟨ - x, 2 y - x ⟩

(as shown in Figure 15.4.6.(b)), over

C_{1}

we have

M = - x = t - 1

and

N = 2 y - x = 2 t - (1 - t) = 3 t - 1 .

Computing the flux across

C_{1} :

\begin{aligned} \int_{C_{1}} \vec{F} \cdot \vec{n} d s & = \int_{C_{1}} (M g^{'} (t) - N f^{'} (t)) d t \\ = \int_{0}^{1} ((t - 1) (1) - (3 t - 1) (- 1)) d t \\ = \int_{0}^{1} (4 t - 2) d t \\ = 0 . \end{aligned}

Over

C_{2},

we have

M = - x = - \cos t

and

N = 2 y - x = 2 \sin t - \cos t .

Thus the flux across

C_{2}

is:

\begin{aligned} \int_{C_{1}} \vec{F} \cdot \vec{n} d s & = \int_{C_{1}} (M g^{'} (t) - N f^{'} (t)) d t \\ = \int_{0}^{π / 2} ((- \cos t) (\cos t) - (2 \sin t - \cos t) (- \sin t)) d t \\ = \int_{0}^{π / 2} (2 \sin^{2} t - \sin t \cos t - \cos^{2} t) d t \\ = π / 4 - 1 / 2 \approx 0.285 . \end{aligned}

We analyze the results of this example below.

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In Example 15.4.5, we saw that the flux across the two curves was the same when the vector field was

{\vec{F}}_{1} = ⟨ y, - x + 1 ⟩ .

This is not a coincidence. We show why they are equal in Example 15.4.23. In short, the reason is this: the divergence of

{\vec{F}}_{1}

is 0, and when

div \vec{F} = 0,

the flux across any two paths with common beginning and ending points will be the same.

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We also saw in the example that the flux across

C_{1}

was 0 when the field was

{\vec{F}}_{2} = ⟨ - x, 2 y - x ⟩ .

Flux measures “how much” of the field crosses the path from left to right (following the conventions established before). Positive flux means most of the field is crossing from left to right; negative flux means most of the field is crossing from right to left; zero flux means the same amount crosses from each side. When we consider Figure 15.4.6.(b), it seems plausible that the same amount of

{\vec{F}}_{2}

was crossing

C_{1}

from left to right as from right to left.

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Subsection 15.4.2 Green’s Theorem

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There is an important connection between the circulation around a closed region

R

and the curl of the vector field inside of

R,

as well as a connection between the flux across the boundary of

R

and the divergence of the field inside

R .

These connections are described by Green’s Theorem and the Divergence Theorem, respectively. We’ll explore each in turn.

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Figure 15.4.7. Introducing Green’s Theorem

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Green’s Theorem states “the counterclockwise circulation around a closed region

R

is equal to the sum of the curls over

R .

”

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Theorem 15.4.8. Green’s Theorem.

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Let

R

be a closed, bounded region of the plane whose boundary

C

is composed of finitely many smooth curves, let

\vec{r} (t)

be a counterclockwise parametrization of

C,

and let

\vec{F} = ⟨ M, N ⟩

where

N_{x}

and

M_{y}

are continuous over

R .

Then

\oint_{C} \vec{F} \cdot d \vec{r} = \iint_{R} curl \vec{F} d A .

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We’ll explore Green’s Theorem through an example.

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Example 15.4.9. Confirming Green’s Theorem.

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Let

\vec{F} = ⟨ - y, x^{2} + 1 ⟩

and let

R

be the region of the plane bounded by the triangle with vertices

(- 1, 0),

(1, 0)

and

(0, 2),

shown in Figure 15.4.10. Verify Green’s Theorem; that is, find the circulation of

\vec{F}

around the boundary of

R

and show that is equal to the double integral of

curl \vec{F}

over

R .

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Figure 15.4.10. The vector field and planar region used in Example 15.4.9

Solution.

The curve

C

that bounds

R

is composed of 3 lines. While we need to traverse the boundary of

R

in a counterclockwise fashion, we may start anywhere we choose. We arbitrarily choose to start at

(- 1, 0),

move to

(1, 0),

etc., with each line parametrized by

{\vec{r}}_{1} (t),

{\vec{r}}_{2} (t)

and

{\vec{r}}_{3} (t),

respectively.

We leave it to the reader to confirm that the following parametrizations of the three lines are accurate:

${\vec{r}}_{1} (t) = ⟨ 2 t - 1, 0 ⟩,$	for $0 \leq t \leq 1,$	with $\vec{r}_{1}^{'} (t) = ⟨ 2, 0 ⟩,$
${\vec{r}}_{2} (t) = ⟨ 1 - t, 2 t ⟩,$	for $0 \leq t \leq 1,$	with $\vec{r}_{2}^{'} (t) = ⟨ - 1, 2 ⟩,$ and
${\vec{r}}_{3} (t) = ⟨ - t, 2 - 2 t ⟩,$	for $0 \leq t \leq 1,$	with $\vec{r}_{3}^{'} (t) = ⟨ - 1, - 2 ⟩ .$

The circulation around

C

is found by summing the flow along each of the sides of the triangle. We again leave it to the reader to confirm the following computations:

\begin{aligned} \int_{C_{1}} \vec{F} \cdot d {\vec{r}}_{1} & = \int_{0}^{1} ⟨ 0, (2 t - 1)^{2} + 1 ⟩ \cdot ⟨ 2, 0 ⟩ d t = 0, \\ \int_{C_{2}} \vec{F} \cdot d {\vec{r}}_{2} & = \int_{0}^{1} ⟨ - 2 t, (1 - t)^{2} + 1 ⟩ \cdot ⟨ - 1, 2 ⟩ d t = 11 / 3, and \\ \int_{C_{3}} \vec{F} \cdot d {\vec{r}}_{3} & = \int_{0}^{1} ⟨ 2 t - 2, t^{2} + 1 ⟩ \cdot ⟨ - 1, - 2 ⟩ d t = - 5 / 3 . \end{aligned}

The circulation is the sum of the flows:

2 .

We confirm Green’s Theorem by computing

\iint_{R} curl \vec{F} d A .

We find

curl \vec{F} = 2 x + 1 .

The region

R

is bounded by the lines

y = 2 x + 2,

y = - 2 x + 2

and

y = 0 .

Integrating with the order

d x d y

is most straightforward, leading to

\int_{0}^{2} \int_{y / 2 - 1}^{1 - y / 2} (2 x + 1) d x d y = \int_{0}^{2} (2 - y) d y = 2,

which matches our previous measurement of circulation.

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Figure 15.4.11. Verifying Green’s Theorem with an example

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Example 15.4.12. Using Green’s Theorem.

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Let

\vec{F} = ⟨ \sin x, \cos y ⟩

and let

R

be the region enclosed by the curve

C

parametrized by

\vec{r} (t) = ⟨ 2 \cos t + \frac{1}{10} \cos (10 t), 2 \sin t + \frac{1}{10} \sin (10 t) ⟩

0 \leq t \leq 2 π,

as shown in Figure 15.4.13. Find the circulation around

C .

Link to full-sized image

Figure 15.4.13. The vector field and planar region used in Example 15.4.12

Solution.

Computing the circulation directly using the line integral looks difficult, as the integrand will include terms like “

\sin (2 \cos t + \frac{1}{10} \cos (10 t)) .

”

Green’s Theorem states that

\oint_{C} \vec{F} \cdot d \vec{r} = \iint_{R} curl \vec{F} d A;

since

curl \vec{F} = 0

in this example, the double integral is simply 0 and hence the circulation is 0.

Since

curl \vec{F} = 0,

we can conclude that the circulation is 0 in two ways. One method is to employ Green’s Theorem as done above. The second way is to recognize that

\vec{F}

is a conservative field, hence there is a function

f (x, y)

wherein

\vec{F} = \nabla f .

Let

A

be any point on the curve

C;

since

C

is closed, we can say that

C

“begins” and “ends” at

A .

By the Fundamental Theorem of Line Integrals,

\oint_{C} \vec{F} d \vec{r} = f (A) - f (A) = 0 .

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Since Green’s Theorem is an important result, it’s worth taking a minute (or 12) to see why it’s true, in the video in Figure 15.4.14.

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Figure 15.4.14. Sketching the proof of Green’s Theorem

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One can use Green’s Theorem to find the area of an enclosed region by integrating along its boundary. Let

C

be a closed curve, enclosing the region

R,

parametrized by

\vec{r} (t) = ⟨ f (t), g (t) ⟩ .

We know the area of

R

is computed by the double integral

\iint_{R} d A,

where the integrand is

1 .

By creating a field

\vec{F}

where

curl \vec{F} = 1,

we can employ Green’s Theorem to compute the area of

R

\oint_{C} \vec{F} \cdot d \vec{r} .

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One is free to choose any field

\vec{F}

to use as long as

curl \vec{F} = 1 .

Common choices are

\vec{F} = ⟨ 0, x ⟩,

\vec{F} = ⟨ - y, 0 ⟩

and

\vec{F} = ⟨ - y / 2, x / 2 ⟩ .

We demonstrate this below.

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Example 15.4.15. Using Green’s Theorem to find area.

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Let

C

be the closed curve parametrized by

\vec{r} (t) = ⟨ t - t^{3}, t^{2} ⟩

- 1 \leq t \leq 1,

enclosing the region

R,

as shown in Figure 15.4.16. Find the area of

R .

Figure 15.4.16. The region

R,

whose area is found in Example 15.4.15

Solution.

We can choose any field

\vec{F},

as long as

curl \vec{F} = 1 .

We choose

\vec{F} = ⟨ - y, 0 ⟩ .

We also confirm (left to the reader) that

\vec{r} (t)

traverses the region

R

in a counterclockwise fashion. Thus

\begin{aligned} Area of R & = \iint_{R} d A \\ = \oint_{C} \vec{F} \cdot d \vec{r} \\ = \int_{- 1}^{1} ⟨ - t^{2}, 0 ⟩ \cdot ⟨ 1 - 3 t^{2}, 2 t ⟩ d t \\ = \int_{- 1}^{1} (- t^{2}) (1 - 3 t^{2}) d t \\ = \frac{8}{15} . \end{aligned}

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Figure 15.4.17. Using Green’s Theorem to find the area under a cycloid

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Another interesting scenario that comes up is the case of multiply-connected regions (as opposed to simply-connected). If a bounded region has a “hole”, its boundary will consist of more than one curve: the outer boundary, as well as the boundary of the hole. Green’s Theorem applies in this situation as well, as the video in Figure 15.4.19 explains.

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Figure 15.4.18. Using Green’s Theorem to derive a polygon area formula

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Figure 15.4.19. Green’s Theorem and multiply-connected regions

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Subsection 15.4.3 The Divergence Theorem

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Green’s Theorem makes a connection between the circulation around a closed region

R

and the sum of the curls over

R .

The Divergence Theorem makes a somewhat “opposite” connection: the total flux across the boundary of

R

is equal to the sum of the divergences over

R .

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Theorem 15.4.20. The Divergence Theorem (in the plane).

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Let

R

be a closed, bounded region of the plane whose boundary

C

is composed of finitely many smooth curves, let

\vec{r} (t)

be a counterclockwise parametrization of

C,

and let

\vec{F} = ⟨ M, N ⟩

where

M_{x}

and

N_{y}

are continuous over

R .

Then

\oint_{C} \vec{F} \cdot \vec{n} d s = \iint_{R} div \vec{F} d A .

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Example 15.4.21. Confirming the Divergence Theorem.

🔗

Let

\vec{F} = ⟨ x - y, x + y ⟩,

let

C

be the circle of radius 2 centered at the origin and define

R

to be the interior of that circle, as shown in Figure 15.4.22. Verify the Divergence Theorem; that is, find the flux across

C

and show it is equal to the double integral of

div \vec{F}

over

R .

Link to full-sized image

Figure 15.4.22. The region

R

used in Example 15.4.21

Solution.

We parametrize the circle in the usual way, with

\vec{r} (t) =

⟨ 2 \cos t, 2 \sin t ⟩,

0 \leq t \leq 2 π .

The flux across

C

\begin{aligned} \oint_{C} \vec{F} \cdot \vec{n} d s & = \oint_{C} (M g^{'} (t) - N f^{'} (t)) d t \\ = \int_{0}^{2 π} ((2 \cos t - 2 \sin t) (2 \cos t) - (2 \cos t + 2 \sin t) (- 2 \sin t)) d t \\ = \int_{0}^{2 π} 4 d t = 8 π . \end{aligned}

We compute the divergence of

\vec{F}

div \vec{F} = M_{x} + N_{y} = 2 .

Since the divergence is constant, we can compute the following double integral easily:

\iint_{R} div \vec{F} d A = \iint_{R} 2 d A = 2 \iint_{R} d A = 2 (area of R) = 8 π,

which matches our previous result.

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Example 15.4.23. Flux when $div \vec{F} = 0$ .

🔗

Let

\vec{F}

be any field where

div \vec{F} = 0,

and let

C_{1}

and

C_{2}

be any two nonintersecting paths, except that each begin at point

A

and end at point

B

(see Figure 15.4.24). Show why the flux across

C_{1}

and

C_{2}

is the same.

Solution.

By referencing Figure 15.4.24, we see we can make a closed path

C

that combines

C_{1}

with

C_{2},

where

C_{2}

is traversed with its opposite orientation. We label the enclosed region

R .

Since

div \vec{F} = 0,

the Divergence Theorem states that

\oint_{C} \vec{F} \cdot \vec{n} d s = \iint_{R} div \vec{F} d A = \iint_{R} 0 d A = 0 .

Using the properties and notation given in Theorem 15.3.10, consider:

Link to full-sized image

Figure 15.4.24. As used in Example 15.4.23, the vector field has a divergence of 0 and the two paths only intersect at their initial and terminal points.

\begin{aligned} 0 & = \oint_{C} \vec{F} \cdot \vec{n} d s \\ = \int_{C_{1}} \vec{F} \cdot \vec{n} d s + \int_{C_{2}^{*}} \vec{F} \cdot \vec{n} d s \end{aligned}

(where

C_{2}^{*}

is the path

C_{2}

traversed with opposite orientation)

\begin{aligned} = \int_{C_{1}} \vec{F} \cdot \vec{n} d s - \int_{C_{2}} \vec{F} \cdot \vec{n} d s . \\ \int_{C_{2}} \vec{F} \cdot \vec{n} d s & = \int_{C_{1}} \vec{F} \cdot \vec{n} d s . \end{aligned}

Thus the flux across each path is equal.

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In this section, we have investigated flow and flux, quantities that measure interactions between a vector field and a planar curve. We can also measure flow along spatial curves, though as mentioned before, it does not make sense to measure flux across spatial curves.

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It does, however, make sense to measure the amount of a vector field that passes across a surface in space — i.e, the flux across a surface. We will study this, though in the next section we first learn about a more powerful way to describe surfaces than using functions of the form