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VECTOR CALCULUS

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.

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VECTOR CALCULUS

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  1. 13 VECTOR CALCULUS

  2. VECTOR CALCULUS 13.8 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.

  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.

  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 12.5 • 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.

  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 12.7 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.

  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.

  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 12.7 with z =g(x, y) = 2 – y, we have the following result.

  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.

  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).

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

  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.

  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.

  47. 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.

  48. 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.

  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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