VECTOR CALCULUS

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VECTOR CALCULUS
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VECTOR CALCULUS
17.8 Stokes Theorem
In this section, we will learn about The Stokes
Theorem and using it to evaluate integrals.
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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 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 (a space curve).

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INTRODUCTION

• The figure shows an oriented surface with unit
  normal vector n.
• The orientation of S induces the positive
  orientation of the boundary curve C.

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INTRODUCTION

• This means that
• If you walk in the positive direction around C
  with your head pointing in the direction of n,
  the surface will always be on your left.

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STOKES THEOREM

• Let
• S be an oriented piecewise-smooth surface
  bounded by a simple, closed, piecewise-smooth
  boundary curve C with positive orientation.
• F be a vector field whose components have
  continuous partial derivatives on an open region
  in that contains S.
• Then,

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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 of S
  of the tangential component of F is equal to the
  surface integral of the normal component of the
  curl of F.

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STOKES THEOREM
Equation 1

• The positively oriented boundary curve of the
  oriented surface S is often written as ?S.
• So, the theorem can be expressed as

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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 is a sort of derivative of
  F).
• The right side involves the values of F only on
  the boundary of S.

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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

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STOKES THEOREM, GREENS THEOREM, FTC

• This is precisely the vector form of Greens
  Theorem given in Equation 12 in Section 16.5
• Thus, we see that Greens Theorem is really a
  special case of Stokes Theorem.

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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
• S is a graph.
• F, S, and C are well behaved.

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STOKES TH.SPECIAL CASE
Proof

• We assume that the equation of S is z g(x,
  y), (x, y) D where
• g has continuous second-order partial
  derivatives.
• D is a simple plane region whose boundary curve
  C1 corresponds to C.

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STOKES TH.SPECIAL CASE
Proof

• If the orientation of S is upward, the positive
  orientation of C corresponds to the positive
  orientation of C1.

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STOKES TH.SPECIAL CASE
Proof

• We are also given that
• F P i Q j R kwhere
  the partial derivatives of P, Q, and R are
  continuous.

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STOKES TH.SPECIAL CASE
Proof

• S is a graph of a function.
• Thus, we can apply Formula 10 in Section 16.7
  with F replaced by curl F.

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STOKES TH.SPECIAL CASE
ProofEquation 2

• The result is
• where the partial derivatives of P, Q, and R are
  evaluated at (x, y, g(x, y)).

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STOKES TH.SPECIAL CASE
Proof

• Suppose
• x x(t) y y(t) a t b
• is a parametric representation of C1.
• Then, a parametric representation of C is x
  x(t) y y(t) z g(x(t), y(t)) a t b

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STOKES TH.SPECIAL CASE
Proof

• This allows us, with the aid of the Chain Rule,
  to evaluate the line integral as follows

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STOKES TH.SPECIAL CASE
Proof

• We have used Greens Theorem in the last step.

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STOKES TH.SPECIAL CASE
Proof

• Next, we use the Chain Rule again, remembering
  that
• P, Q, and R are functions of x, y, and z.
• z is itself a function of x and y.

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STOKES TH.SPECIAL CASE
Proof

• Thus, we get

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STOKES TH.SPECIAL CASE
Proof

• Four terms in that double integral cancel.
• The remaining six can be arranged to coincide
  with the right side of Equation 2.
• Hence,

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STOKES THEOREM
Example 1

• Evaluate where
• F(x, y, z) y2 i x j z2 k
• 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.)

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STOKES THEOREM
Example 1

• The curve C (an ellipse) is shown here.
• could be evaluated directly.
• However, its easier to use Stokes Theorem.

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STOKES THEOREM
Example 1

• We first compute

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STOKES THEOREM
Example 1

• There are many surfaces with boundary C.
• The most convenient choice, though, is the
  elliptical region S in the plane y z 2 that
  is bounded by C.
• If we orient S upward, C has the induced
  positive orientation.

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STOKES THEOREM
Example 1

• The projection D of S on the xy-plane is the
  disk x2 y2 1.
• So, using Equation 10 in Section 16.7 with z
  g(x, y) 2 y, we have the following result.

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STOKES THEOREM
Example 1
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STOKES THEOREM
Example 2

• Use Stokes Theorem to compute where
• F(x, y, z) xz i yz j xy k
• S is the part of the sphere x2 y2 z2 4
  that lies inside the cylinder x2 y2 1 and
  above the xy-plane.

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STOKES THEOREM
Example 2

• To find the boundary curve C, we solve x2
  y2 z2 4 and x2 y2 1
• Subtracting, we get z2 3.
• So, (since z gt 0).

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STOKES THEOREM
Example 2

• So, C is the circle given by x2 y2 1,

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STOKES THEOREM
Example 2

• A vector equation of C is r(t) cos t i sin
  t j k 0 t 2p
• Therefore, r(t) sin t i cos t j
• Also, we have

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STOKES THEOREM
Example 2

• Thus, by Stokes Theorem,

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STOKES THEOREM

• Note that, in Example 2, we computed a surface
  integral simply by knowing the values of F on
  the boundary curve C.
• This means that
• If we have another oriented surface with the
  same boundary curve C, we get exactly the same
  value for the surface integral!

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STOKES THEOREM
Equation 3

• 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.

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CURL VECTOR

• We now use Stokes Theorem to throw some light on
  the meaning of the curl vector.
• Suppose that C is an oriented closed curve and v
  represents the velocity field in fluid flow.

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CURL VECTOR

• Consider the line integral and recall that v
  T is the component of v in the direction of the
  unit tangent vector T.
• This means that the closer the direction of v is
  to the direction of T, the larger the value of v
  T.

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CIRCULATION

• Thus, is a measure of the tendency
  of the fluid to move around C.
• It is called the circulation of v around C.

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CURL VECTOR

• Now, let P0(x0, y0, z0) be a point in the
  fluid.
• Sa be a small disk with radius a and center P0.
• Then, (curl F)(P) (curl F)(P0) for all points
  P on Sa because 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

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CURL VECTOR
Equation 4

• The approximation becomes better as a ? 0.
• Thus, we have

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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 16.5
• If curl F 0 on all of , then F is
  conservative.

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CLOSED CURVES

• From Theorems 3 and 4 in Section 16.3, we know
  that F is conservative if for
  every closed path C.
• Given C, suppose we can find an orientable
  surface S whose boundary is C.
• This can be done, but the proof requires
  advanced techniques.

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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.

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CLOSED CURVES

• Adding these integrals, we obtain for any
  closed curve C.