slide1 n.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
VECTOR CALCULUS PowerPoint Presentation
Download Presentation
VECTOR CALCULUS

Loading in 2 Seconds...

play fullscreen
1 / 50

VECTOR CALCULUS - PowerPoint PPT Presentation


  • 114 Views
  • Uploaded on

13. VECTOR CALCULUS. VECTOR CALCULUS. 13.8 Stokes’ Theorem. In this section, we will learn about: The Stokes’ Theorem and using it to evaluate integrals. STOKES’ VS. GREEN’S THEOREM. Stokes’ Theorem can be regarded as a higher-dimensional version of Green’s Theorem.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'VECTOR CALCULUS' - deva


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
slide1

13

VECTOR CALCULUS

slide2

VECTOR CALCULUS

13.8

Stokes’ Theorem

In this section, we will learn about:

The Stokes’ Theorem and

using it to evaluate integrals.

stokes vs green s theorem
STOKES’ VS. GREEN’S THEOREM
  • Stokes’ Theorem can be regarded as a higher-dimensional version of Green’s Theorem.
    • 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 (a space curve).
introduction
INTRODUCTION
  • The figure shows an oriented surface with unit normal vector n.
    • The orientation of Sinduces the positive orientation of the boundary curve C.
introduction1
INTRODUCTION
  • This means that:
    • If you walk in the positive direction around Cwith your head pointing in the direction of n, the surface will always be on your left.
stokes theorem
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,
stokes theorem1
STOKES’ THEOREM
  • The theorem is named after the Irish mathematical physicist Sir George Stokes (1819–1903).
    • What we call Stokes’ Theorem was actually discovered by the Scottish physicist Sir William Thomson (1824–1907, known as Lord Kelvin).
    • Stokes learned of it in a letter from Thomson in 1850.
stokes theorem2
STOKES’ THEOREM
  • Thus, Stokes’ Theorem says:
    • The line integral around the boundary curve of Sof the tangential component of F is equal to the surface integral of the normal component of the curl of F.
stokes theorem3
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:
stokes theorem green s theorem ftc
STOKES’ THEOREM, GREEN’S THEOREM, & FTC
  • There is an analogy among Stokes’ Theorem, Green’s 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 boundaryof S.
stokes theorem green s theorem ftc1
STOKES’ THEOREM, GREEN’S THEOREM, & FTC
  • In fact, consider the special case where the surface S:
    • Is flat.
    • Lies in the xy-plane with upward orientation.
stokes theorem green s theorem ftc2
STOKES’ THEOREM, GREEN’S THEOREM, & FTC
  • Then,
    • The unit normal is k.
    • The surface integral becomes a double integral.
    • Stokes’ Theorem becomes:
stokes theorem green s theorem ftc3
STOKES’ THEOREM, GREEN’S THEOREM, & FTC
  • This is precisely the vector form of Green’s Theorem given in Equation 12 in Section 12.5
    • Thus, we see that Green’s Theorem is really a special case of Stokes’ Theorem.
stokes theorem4
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.
stokes th special case
STOKES’ TH.—SPECIAL CASE

Proof

  • We assume that the equation of Sis: z = g(x, y), (x, y) Dwhere:
    • g has continuous second-order partial derivatives.
    • D is a simple plane region whose boundary curve C1 corresponds to C.
stokes th special case1
STOKES’ TH.—SPECIAL CASE

Proof

  • If the orientation of S is upward, the positive orientation of C corresponds to the positive orientation of C1.
stokes th special case2
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.
stokes th special case3
STOKES’ TH.—SPECIAL CASE

Proof

  • S is a graph of a function.
  • Thus, we can apply Formula 10 in Section 12.7 with F replaced by curl F.
stokes th special case4
STOKES’ TH.—SPECIAL CASE

Proof—Equation 2

  • The result is:
  • where the partial derivatives of P, Q, and Rare evaluated at (x, y, g(x, y)).
stokes th special case5
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 Cis: x =x(t) y =y(t) z =g(x(t), y(t)) a ≤t ≤b
stokes th special case6
STOKES’ TH.—SPECIAL CASE

Proof

  • This allows us, with the aid of the Chain Rule, to evaluate the line integral as follows:
stokes th special case7
STOKES’ TH.—SPECIAL CASE

Proof

  • We have used Green’s Theorem in the last step.
stokes th special case8
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.
stokes th special case9
STOKES’ TH.—SPECIAL CASE

Proof

  • Thus, we get:
stokes th special case10
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,
stokes theorem5
STOKES’ THEOREM

Example 1

  • Evaluate where:
    • F(x, y, z) = –y2i + x j + z2k
    • 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.)
stokes theorem6
STOKES’ THEOREM

Example 1

  • The curve C (an ellipse) is shown here.
    • could be evaluated directly.
    • However, it’s easier to use Stokes’ Theorem.
stokes theorem7
STOKES’ THEOREM

Example 1

  • We first compute:
stokes theorem8
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.
stokes theorem9
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 12.7 with z =g(x, y) = 2 – y, we have the following result.
stokes theorem11
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.
stokes theorem12
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 > 0).
stokes theorem13
STOKES’ THEOREM

Example 2

  • So, C is the circle given by: x2 + y2 = 1,
stokes theorem14
STOKES’ THEOREM

Example 2

  • A vector equation of C is:r(t) = cos t i + sin t j + k 0 ≤t ≤ 2π
    • Therefore, r’(t) =–sin t i + cos t j
  • Also, we have:
stokes theorem15
STOKES’ THEOREM

Example 2

  • Thus, by Stokes’ Theorem,
stokes theorem16
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!
stokes theorem17
STOKES’ THEOREM

Equation 3

  • In general, if S1 and S2 are oriented surfaces with the same oriented boundary curve Cand 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.
curl vector
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.
curl vector1
CURL VECTOR
  • Consider the line integral and recall that v ∙T is the component of vin 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.
circulation
CIRCULATION
  • Thus, is a measure of the tendency of the fluid to move around C.
    • It iscalled the circulation of v around C.
curl vector2
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.
curl vector3
CURL VECTOR
  • Thus, by Stokes’ Theorem, we get the following approximation to the circulation around the boundary circle Ca:
curl vector4
CURL VECTOR

Equation 4

  • The approximation becomes better as a→ 0.
  • Thus, we have:
curl circulation
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.
curl circulation1
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.
closed curves
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.
closed curves1
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 orientable surface S whose boundary is C.
    • This can be done, but the proof requires advanced techniques.
closed curves2
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.
closed curves3
CLOSED CURVES
  • Adding these integrals, we obtain: for any closed curve C.