1. SECTION 13.8 STOKES ’ THEOREM

2. P2P213.8 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).

3. P3P313.8 INTRODUCTION  Figure 1 shows an oriented surface with unit normal vector n. The orientation of S induces the positive orientation of the boundary curve C shown in the figure.

4. P4P413.8 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.

5. P5P513.8 STOKES ’ THEOREM  Let S be an oriented piecewise-smooth surface bounded by a simple, closed, piecewise-smooth boundary curve C with positive orientation. Let F be a vector field whose components have continuous partial derivatives on an open region in that contains S. Then,

6. P6P613.8 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.

7. P7P713.8 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.

8. P8P813.8 STOKES ’ THEOREM  The positively oriented boundary curve of the oriented surface S is often written as ∂S.  So, the theorem can be expressed as:

9. P9P913.8 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 boundary of S.

10. P1013.8 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.

11. P1113.8 STOKES ’ THEOREM, GREEN ’ S THEOREM, & FTC  Then, The unit normal is k. The surface integral becomes a double integral. Stokes ’ Theorem becomes:

12. P1213.8 STOKES ’ THEOREM, GREEN ’ S THEOREM, & FTC  This is precisely the vector form of Green ’ s Theorem given in Equation 12 in Section 13.5 Thus, we see that Green ’ s Theorem is really a special case of Stokes ’ Theorem.

13. P1313.8 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.

14. P1413.8 PROOF OF A SPECIAL CASE OF STOKES ’ THEOREM  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 C 1 corresponds to C.

15. P1513.8 PROOF OF A SPECIAL CASE OF STOKES ’ THEOREM  If the orientation of S is upward, the positive orientation of C corresponds to the positive orientation of C 1.

16. P1613.8 PROOF OF A SPECIAL CASE OF STOKES ’ THEOREM  We are also given that: F = P i + Q j + R k where the partial derivatives of P, Q, and R are continuous.  S is a graph of a function.  Thus, we can apply Formula 10 in Section 13.7 with F replaced by curl F.

17. P1713.8 PROOF OF A SPECIAL CASE OF STOKES ’ THEOREM  The result is: where the partial derivatives of P, Q, and R are evaluated at (x, y, g(x, y)).

18. P1813.8 PROOF OF A SPECIAL CASE OF STOKES ’ THEOREM  Suppose x = x(t) y = y(t) a ≤ t ≤ b is a parametric representation of C 1. Then, a parametric representation of C is: x = x(t) y = y(t) z = g(x(t), y(t)) a ≤ t ≤ b

19. P1913.8 PROOF OF A SPECIAL CASE OF STOKES ’ THEOREM  This allows us, with the aid of the Chain Rule, to evaluate the line integral as follows:

20. P2013.8 PROOF OF A SPECIAL CASE OF STOKES ’ THEOREM We have used Green ’ s Theorem in the last step.

21. P2113.8 PROOF OF A SPECIAL CASE OF STOKES ’ THEOREM  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.

22. P2213.8 PROOF OF A SPECIAL CASE OF STOKES ’ THEOREM  Thus, we get:

23. P2313.8 PROOF OF A SPECIAL CASE OF STOKES ’ THEOREM  Four terms in that double integral cancel.  The remaining six can be arranged to coincide with the right side of Equation 2. Hence,

24. P2413.8 Example 1  Evaluate where: F(x, y, z) = – y 2 i + x j + z 2 k C is the curve of intersection of the plane y + z = 2 and the cylinder x 2 + y 2 = 1. (Orient C to be counterclockwise when viewed from above.)

25. P2513.8 Example 1 SOLUTION  The curve C (an ellipse) is shown in Figure 3. could be evaluated directly. However, it ’ s easier to use Stokes ’ Theorem.

26. P2613.8 Example 1 SOLUTION  We first compute:

27. P2713.8 Example 1 SOLUTION  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.

28. P2813.8 Example 1 SOLUTION  The projection D of S on the xy-plane is the disk x 2 + y 2 ≤ 1. So, using Equation 10 in Section 13.7 with z = g(x, y) = 2 – y, we have the following result.

29. P2913.8 Example 1 SOLUTION

30. P3013.8 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 x 2 + y 2 + z 2 = 4 that lies inside the cylinder x 2 + y 2 =1 and above the xy-plane.

31. P3113.8 Example 2 SOLUTION  To find the boundary curve C, we solve: x 2 + y 2 + z 2 = 4 and x 2 + y 2 = 1 Subtracting, we get z 2 = 3. So, (since z > 0).  So, C is the circle given by: x 2 + y 2 = 1,

32. P3213.8 Example 2 SOLUTION  A vector equation of C is: r(t) = cos t i + sin t j + k 0 ≤ t ≤ 2  Therefore, r ’ (t) = – sin t i + cos t j  Also, we have:

33. P3313.8 Example 2 SOLUTION  Thus, by Stokes ’ Theorem,

34. P3413.8 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!

35. P3513.8 STOKES ’ THEOREM  In general, if S 1 and S 2 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.

36. P3613.8 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.

37. P3713.8 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.

38. P3813.8 CIRCULATION  Thus, is a measure of the tendency of the fluid to move around C. It is called the circulation of v around C. See Figure 5.

39. P3913.8 CURL VECTOR  Now, let P 0 (x 0, y 0, z 0 ) be a point in the fluid.  S a be a small disk with radius a and center P 0. Then, (curl F)(P) ≈ (curl F)(P 0 ) for all points P on S a because curl F is continuous.

40. P4013.8 CURL VECTOR  Thus, by Stokes ’ Theorem, we get the following approximation to the circulation around the boundary circle C a :

41. P4113.8 CURL VECTOR  The approximation becomes better as a → 0.  Thus, we have:

42. P4213.8 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.

43. P4313.8 CURL & CIRCULATION  The paddle wheel rotates fastest when its axis is parallel to curl v.

44. P4413.8 CLOSED CURVES  Finally, we mention that Stokes ’ Theorem can be used to prove Theorem 4 in Section 13.5: If curl F = 0 on all of, then F is conservative.

45. P4513.8 CLOSED CURVES  From Theorems 3 and 4 in Section 13.3, 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.

46. P4613.8 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.

47. P4713.8 CLOSED CURVES  Adding these integrals, we obtain: for any closed curve C.