chap 13 econ
TRANSCRIPT
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VECTOR CALCULUS
13
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13.8
Stokes Theorem
In this section, we will learn about:
The Stokes Theorem and
using it to evaluate integrals.
VECTOR CALCULUS
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STOKES VS. GREENS THEOREM
Stokes Theorem can be regarded as
a higher-dimensional version of Greens
Theorem.
Greens Theorem relates a double integral over
a plane region Dto a line integral around its planeboundary curve.
Stokes Theorem relates a surface integral over
a surface Sto a line integral around the boundarycurve ofS(a space curve).
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INTRODUCTION
The figure shows an oriented surface with
unit normal vectorn.
The orientation ofSinduces the positiveorientation of the
boundary curve C.
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INTRODUCTION
This means that:
If you walk in the positive direction around Cwith your head pointing in the direction ofn,
the surface will always be on your left.
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STOKES THEOREM
Let:
Sbe an oriented piecewise-smooth surfacebounded by a simple, closed, piecewise-smooth
boundary curve Cwith positive orientation.
F be a vector field whose components havecontinuous partial derivatives on an open region
in that contains S.
Then,
3
curlC
S
d d F r F S
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STOKES THEOREM
The theorem is named after the Irish
mathematical physicist Sir George Stokes
(18191903).
What we call Stokes Theorem was actually
discovered by the Scottish physicist Sir William
Thomson (18241907, known as Lord Kelvin).
Stokes learned of it in a letter from Thomson in 1850.
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STOKES THEOREM
Thus, Stokes Theorem says:
The line integral around the boundary curve ofSof the tangential component ofF is equal to the surfaceintegral of the normal component of the curl ofF.
and
curl curl
C C
S S
d ds
d d
F r F T
F S F n S
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STOKES THEOREM
The positively oriented boundary curve of
the oriented surface Sis often written as S.
So,the theorem can be expressed as:
curlS
S
d d
F S F r
Equation 1
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STOKES THEOREM, GREENS THEOREM, & FTC
There is an analogy among Stokes Theorem,
Greens Theorem, and the Fundamental
Theorem of Calculus (FTC).
As before, there is an integral involving derivatives
on the left side of Equation 1 (recall that curl F isa sort of derivative ofF).
The right side involves the values ofF only onthe boundaryofS.
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STOKES THEOREM, GREENS THEOREM, & FTC
In fact, consider the special case
where the surface S:
Is flat.
Lies in the xy-plane with upward orientation.
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STOKES THEOREM, GREENS THEOREM, & FTC
Then,
The unit normal is k.
The surface integral becomes a double integral.
Stokes Theorem becomes:
curl curlC
S S
d d dA F r F S F k
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STOKES THEOREM
Stokes Theorem is too difficult for us to
prove in its full generality.
Still, we can give a proof when:
Sis a graph.
F, S, and Care well behaved.
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STOKES TH.SPECIAL CASE
We assume that the equation ofS
is:
z= g(x, y), (x, y) D
where:
ghas continuous second-order partial derivatives.
Dis a simple plane region whose boundary curveC1 corresponds to C.
Proof
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If the orientation ofSis upward, the positive
orientation ofCcorresponds to the positive
orientation ofC1.
ProofSTOKES TH.SPECIAL CASE
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STOKES TH.SPECIAL CASE
We are also given that:
F = Pi + Qj + Rk
where the partial derivatives of
P, Q, and Rare continuous.
Proof
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STOKES TH.SPECIAL CASE
Sis a graph of a function.
Thus, we can apply Formula 10 in
Section 12.7 with F replaced by curl F.
Proof
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STOKES TH.SPECIAL CASE
Suppose
x =x(t) y =y(t) atb
is a parametric representation ofC1.
Then, a parametric representation ofCis:
x =x(t) y =y(t) z =g(x(t), y(t)) a
t
b
Proof
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STOKES TH.SPECIAL CASE
Four terms in that double integral cancel.
The remaining six can be arranged to
coincide with the right side of Equation 2.
Hence,
curlCS
d d F r F S
Proof
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STOKES THEOREM
We first compute:
2 2
curl 1 2yx y z
y x z
i j k
F k
Example 1
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STOKES THEOREM
There are many surfaces with
boundary C.
The most convenientchoice, though, isthe elliptical region Sin the plane y+ z= 2that is bounded by C.
If we orient Supward,Chas the inducedpositive orientation.
Example 1
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STOKES THEOREM
The projection DofSon the xy-plane
is the disk x2 + y2 1.
So, using Equation 10in Section 12.7 with
z =g(x, y) = 2y,we have the following
result.
Example 1
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STOKES THEOREM
2 1
0 0
12 32
00
21 2
2 30
1
2
curl 1 2
1 2 sin
2 sin2 3
sin
2 0
CS D
d d y dA
r r dr d
r rd
d
F r F S
Example 1
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STOKES THEOREM
Use Stokes Theorem to compute
where:
F(x, y, z) = xzi + yzj + xyk
Sis the part of
the spherex2 + y2 + z2 = 4that lies inside
the cylinder
x2 + y2 =1and above
the xy-plane.
curlS
d F SExample 2
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STOKES THEOREM
A vector equation ofCis:
r(t) = cos ti + sin tj + k 0 t 2
Therefore, r(t) =sin ti + cos tj
Also, we have:
3
Example 2
3 cos 3 sin cos sint t t t t F r i j k
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STOKES THEOREM
Thus, by Stokes Theorem,
2
0
2
0
2
0
curl
( ( )) '( )
3 cos sin 3 sin cos
3 0 0
CS
d d
t t dt
t t t t dt
dt
F S F r
F r r
Example 2
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STOKES THEOREM
In general, ifS1 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.
1 2
curl curlC
S S
d d d F S F r F S
Equation 3
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CIRCULATION
Thus, is a measure of the tendency
of the fluid to move around C.
It iscalled the circulation ofv around C.
C
d v r
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CURL VECTOR
Now, let:
P0(x0, y0, z0) be a point in the fluid.
Sabe a small disk with radius aand centerP0.
Then, (curl F)(P) (curl F)(P0) for all pointsPon Sabecause curl F is continuous.
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CURL VECTOR
Thus, by Stokes Theorem, we get
the following approximation to the circulation
around the boundary circle Ca:
0 0
2
0 0
curl curl
curl
curl
a
a a
a
CS S
S
d d dS
P P dS
P P a
v r v S v n
v n
v n
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CURL VECTOR
The approximation becomes better as a 0.
Thus, we have:
0 0 201
curl limaCa
P P da
v n v r
Equation 4
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CURL & CIRCULATION
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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CURL & CIRCULATION
Imagine a tiny paddle wheel placed in
the fluid at a point P.
The paddle wheel
rotates fastest
when its axis is
parallel to curl v.
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CLOSED CURVES
Finally, we mention that Stokes Theorem
can be used to prove Theorem 4 in
Section 12.5:
If curl F = 0 on all of , then F is conservative.3
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CLOSED CURVES
From Theorems 3 and 4 in Section 12.3,
we know that F is conservative if
for every closed path C.
Given C, suppose we can find an orientablesurface Swhose boundary is C.
This can be done, but the proof requiresadvanced techniques.
0C
d F r
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CLOSED CURVES
Then, Stokes Theorem gives:
A curve that is not simple can be broken into
a number of simple curves. The integrals around these curves are all 0.
curl 0 0C
S S
d d d F r F S S
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CLOSED CURVES
Adding these integrals,
we obtain:
for any closed curve C.
0C
d F r