The Theorem of Stokes in the Plane

Section 11.3 The Theorem of Stokes in the Plane

Given a vector field

f = (f_{1}, f_{2}),

Green’s theorem asserts that

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\iint_{D} (\frac{\partial}{\partial x_{1}} f_{2} - \frac{\partial}{\partial x_{2}} f_{1}) d x_{1} d x_{2} = \int_{\partial D} f_{1} d x_{1} + f_{2} d x_{2} .

As we saw in Section 7.4 the line integral on the right hand side represents the circulation of the vector field

f

along the curve

\partial D

in the positive direction. In Remark 8.17 we called the integrand on the left hand side the curl of a plane vector field, so we set

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curl f := \frac{\partial}{\partial x_{1}} f_{2} - \frac{\partial}{\partial x_{2}} f_{1}

in this section. With this we can reformulate Green’s theorem as follows.

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Theorem 11.7. Theorem of Stokes.

f

is a smooth vector field defined on a piecewise smooth domain

D

then

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\begin{matrix} (11.3) & \int_{D} curl f (x) d x = \int_{\partial D} f \cdot τ d s, \end{matrix}

where

τ

is the positive unit tangent to

\partial D .

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We next want to obtain a physical interpretation of

curl f .

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Observation 11.8. Physical interpretation of curl.

Let

f

be the velocity of a fluid or a gas in a region

D .

Fix a point

a = (a_{1}, a_{2})

D,

and denote by

B_{δ}

a disc centred at

a

with radius

δ .

Then

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\int_{\partial B_{δ}} f \cdot τ d s

is the net flow of

f

along

\partial B_{δ} .

In other words it is the circulation of the field along

\partial B_{δ} .

By the theorem of Stokes

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\frac{1}{area (B_{δ})} \int_{B_{δ}} curl f d x = \frac{1}{area (B_{δ})} \int_{\partial B_{δ}} f \cdot τ d s .

The above is a density of the circulation of

f

B_{δ} .

If we shrink

B_{δ}

a

we get (see Lemma 11.6)

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lim_{δ \to 0} \frac{1}{area (B_{δ})} \int_{B_{δ}} curl f d x = curl f (a) .

Hence

curl f

is a measure for the circulation of the vector field

f

a .

If we fix a little paddle wheel at

a

then

curl f (a)

tells us how it turns. If

curl f (a) > 0

it turns in the counterclockwise direction, if

curl f (a) < 0

it turns in the clockwise direction, and if

curl f (a) = 0

it does not turn at all.

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Example 11.9.

Let

f = (y, 0)

for

(x, y) \in R^{2} .

Then the vector field looks as shown in Figure 11.10. A simple computation shows that

curl f = - 1,

so a little paddle wheel would turn in the clockwise direction. This is clear as the current is different on the top and the bottom.

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figure 1

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Figure 11.10. The vector field

f (x, y) := (y, 0)

R^{2} .

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Example 11.11.

Let

f = (- y, x)

for

(x, y) \in R^{2} .

Then the vector field looks as shown in Figure 11.12. A simple computation shows that

curl f = 2,

so a little paddle wheel would turn in the counterclockwise direction.

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Figure 11.12. The vector field

f (x, y) := (- y, x)

R^{2} .

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As the vector field circles around the origin it seems `obvious’ that the curl is positive. Note however, that the curl is a local property of the vector field, and that a global picture can be quite deceptive as the following example shows!

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Example 11.13.

Let

f (x, y) := (\frac{- y}{x^{2} + y^{2}}, \frac{x}{x^{2} + y^{2}})

be the vector field studied in Example 8.12. It looks as shown in Figure 8.13. It follows from the computations there that

curl f = 0

except at the origin, where

f

is not defined. On the other hand the field looks at first quite similar to the one from the previous example, where the curl was positive.

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