16.8

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16.8 Stokes’ Theorem

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Stokes’ Theorem

Stokes’ Theorem can be regarded as a higher-dimensional version of Green’s Theorem.

Whereas Green’s Theorem relates a double integral over a plane region D to a line integral around its plane

boundary curve, Stokes’ Theorem relates a surface integral over a surface S to a line integral around the boundary

curve of S (which is a space curve).

Figure 1 shows an oriented surface with unit normal

vector n.

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Stokes’ Theorem

The orientation of S induces the positive orientation of the boundary curve C shown in the figure.

This means that if you walk in the positive direction around C with your head pointing in the direction of n, then the

surface will always be on your left.

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Stokes’ Theorem

Since

∫

_C ^F^ ^{dr =}

∫

_C ^F^ T ds and curl F  dS = curl F  n dS Stokes’ Theorem says that the line integral around the

boundary curve of S of the tangential component of F is equal to the surface integral over S of the normal

component of the curl of F.

The positively oriented boundary curve of the oriented surface S is often written as ∂S, so Stokes’ Theorem can be expressed as

curl F  dS = F  dr

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Stokes’ Theorem

There is an analogy among Stokes’ Theorem, Green’s Theorem, and the Fundamental Theorem of Calculus.

As before, there is an integral involving derivatives on the left side of Equation 1 (we know that curl F is a sort of

derivative of F) and the right side involves the values of F only on the boundary of S.

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Stokes’ Theorem

In fact, in the special case where the surface S is flat and lies in the xy-plane with upward orientation, the unit normal is k, the surface integral becomes a double integral, and Stokes’ Theorem becomes

∫

_C ^F^ dr = curl F  dS = (curl F)  k dA

This is precisely the vector form of Green’s Theorem.

Thus we see that Green’s Theorem is really a special case of Stokes’ Theorem.

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Example 1

Evaluate

∫

_C F  dr, where F(x, y, z) = –y²i + x j + z²k and C is the curve of intersection of the plane y + z = 2 and the cylinder x² + y² = 1. (Orient C to be counterclockwise when viewed from above.)

Solution:

The curve C (an ellipse) is shown in Figure 3.

Figure 3

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Example 1 – Solution

Although

∫

_C F  dr could be evaluated directly, it’s easier to use Stokes’ Theorem.

We first compute

Although there are many surfaces with boundary C, the most convenient choice is the elliptical region S in the plane y + z = 2 that is bounded by C.

cont’d

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Example 1 – Solution

If we orient S upward, then C has the induced positive orientation.

The projection D of S onto the xy-plane is the disk x² + y² ≤ 1 and so using equation

with z = g(x, y) = 2 – y, we have

cont’d

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Example 1 – Solution

_cont’d

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Stokes’ Theorem

In general, if S₁ and S₂ are oriented surfaces with the same oriented boundary curve C and both satisfy the hypotheses of Stokes’ Theorem, then

This fact is useful when it is difficult to integrate over one surface but easy to integrate over the other.

We now use Stokes’ Theorem to throw some light on the meaning of the curl vector.

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Stokes’ Theorem

Suppose that C is an oriented closed curve and v represents the velocity field in fluid flow.

Consider the line integral

∫

_C ^v^ ^{dr =}

∫

_C ^v^ ^{T ds}

and we know that v  T is the component of v in the direction of the unit tangent vector T.

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Stokes’ Theorem

This means that the closer the direction of v is to the direction of T, the larger the value of v  T.

Thus

∫

_C ^v^ dr is a measure of the tendency of the fluid to move around C and is called the circulation of v around C.

(See Figure 5.)

Figure 5

(a) ∫_C v  dr > 0, positive circulation (b) ∫_C v  dr < 0, negative circulation

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Stokes’ Theorem

Now let P₀(x₀, y₀, z₀) be a point in the fluid and let S_a be a small disk with radius a and center P₀.

Then (curl F)(P) ≈ (curl F)(P₀) for all points P on S_a because curl F is continuous.

Thus, by Stokes’ Theorem, we get the following

approximation to the circulation around the boundary circle C_a:

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Stokes’ Theorem

This approximation becomes better as a → 0 and we have

Equation 4 gives the relationship between the curl and the circulation.

It shows that curl v  n is a measure of the rotating effect of the fluid about the axis n.

The curling effect is greatest about the axis parallel to curl v.

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Stokes’ Theorem

We know that F is conservative if

∫

_C F  dr = 0 for every

closed path C. Given C, suppose we can find an orientable surface S whose boundary is C.

Then Stokes’ Theorem gives

A curve that is not simple can be broken into a number of simple curves, and the integrals around these simple

curves are all 0.

Adding these integrals, we obtain

∫

_C F  dr = 0 for any closed curve C.