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VECTOR CALCULUS

17. VECTOR CALCULUS. VECTOR CALCULUS. 17.8 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.

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VECTOR CALCULUS

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  1. 17 VECTOR CALCULUS

  2. VECTOR CALCULUS 17.8 Stokes’ Theorem • In this section, we will learn about: • The Stokes’ Theorem and • using it to evaluate integrals.

  3. 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).

  4. INTRODUCTION • The figure shows an oriented surface with unit normal vector n. • The orientation of Sinduces the positive orientation of the boundary curve C. Fig. 17.8.1, p. 1129

  5. 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. Fig. 17.8.1, p. 1129

  6. 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,

  7. 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.

  8. 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.

  9. 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:

  10. 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.

  11. 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.

  12. STOKES’ THEOREM, GREEN’S THEOREM, & FTC • Then, • The unit normal is k. • The surface integral becomes a double integral. • Stokes’ Theorem becomes:

  13. STOKES’ THEOREM, GREEN’S THEOREM, & FTC • This is precisely the vector form of Green’s Theorem given in Equation 12 in Section 17.5 • Thus, we see that Green’s Theorem is really a special case of Stokes’ Theorem.

  14. 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.

  15. 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.

  16. STOKES’ TH.—SPECIAL CASE Proof • If the orientation of S is upward, the positive orientation of C corresponds to the positive orientation of C1. Fig. 16.8.2, p. 1129

  17. 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.

  18. STOKES’ TH.—SPECIAL CASE Proof • S is a graph of a function. • Thus, we can apply Formula 10 in Section 17.7 with F replaced by curl F.

  19. 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)).

  20. 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

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

  22. STOKES’ TH.—SPECIAL CASE Proof • We have used Green’s Theorem in the last step.

  23. 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.

  24. STOKES’ TH.—SPECIAL CASE Proof • Thus, we get:

  25. 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,

  26. 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.)

  27. STOKES’ THEOREM Example 1 • The curve C (an ellipse) is shown here. • could be evaluated directly. • However, it’s easier to use Stokes’ Theorem. Fig. 17.8.3, p. 1131

  28. STOKES’ THEOREM Example 1 • We first compute:

  29. 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. Fig. 17.8.3, p. 1131

  30. 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 17.7 with z =g(x, y) = 2 – y, we have the following result. Fig. 17.8.3, p. 1131

  31. STOKES’ THEOREM Example 1

  32. 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. Fig. 17.8.4, p. 1131

  33. 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). Fig. 17.8.4, p. 1131

  34. STOKES’ THEOREM Example 2 • So, C is the circle given by: x2 + y2 = 1, Fig. 17.8.4, p. 1131

  35. 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:

  36. STOKES’ THEOREM Example 2 • Thus, by Stokes’ Theorem,

  37. 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!

  38. 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.

  39. 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.

  40. 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.

  41. CIRCULATION • Thus, is a measure of the tendency of the fluid to move around C. • It iscalled the circulation of v around C. Fig. 17.8.5, p. 1132

  42. 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.

  43. CURL VECTOR • Thus, by Stokes’ Theorem, we get the following approximation to the circulation around the boundary circle Ca:

  44. CURL VECTOR Equation 4 • The approximation becomes better as a→ 0. • Thus, we have:

  45. 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.

  46. 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. Fig. 17.8.6, p. 1132

  47. 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.

  48. CLOSED CURVES • From Theorems 3 and 4 in Section 17.3, 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.

  49. 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.

  50. CLOSED CURVES • Adding these integrals, we obtain: for any closed curve C.

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