VECTOR CALCULUS 17

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

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 S induces the positive orientation of the boundary curve C. Fig , p. 1129

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

9 STOKES’ THEOREM The positively oriented boundary curve of the oriented surface S is often written as ∂S. So, the theorem can be expressed as: Equation 1

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 boundary of 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 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. Proof

16 If the orientation of S is upward, the positive orientation of C corresponds to the positive orientation of C 1. Proof STOKES’ TH.—SPECIAL CASE Fig , p. 1129

17 STOKES’ TH.—SPECIAL CASE We are also given that: F = P i + Q j + R k where the partial derivatives of P, Q, and R are continuous. Proof

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

19 STOKES’ TH.—SPECIAL CASE The result is: where the partial derivatives of P, Q, and R are evaluated at (x, y, g(x, y)). Proof—Equation 2

20 STOKES’ TH.—SPECIAL CASE 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 Proof

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

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

23 STOKES’ TH.—SPECIAL CASE 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. Proof

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

25 STOKES’ TH.—SPECIAL CASE Four terms in that double integral cancel. The remaining six can be arranged to coincide with the right side of Equation 2.  Hence, Proof

26 STOKES’ THEOREM 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.) Example 1

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

28 STOKES’ THEOREM We first compute: Example 1

29 STOKES’ THEOREM 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. Example 1 Fig , p. 1131

30 STOKES’ THEOREM The projection D of S on the xy-plane is the disk x 2 + y 2 ≤ 1.  So, using Equation 10 in Section 17.7 with z = g(x, y) = 2 – y, we have the following result. Example 1 Fig , p. 1131

31 STOKES’ THEOREM Example 1

32 STOKES’ THEOREM 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. Example 2 Fig , p. 1131

33 STOKES’ THEOREM 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). Example 2 Fig , p. 1131

34 STOKES’ THEOREM So, C is the circle given by: x 2 + y 2 = 1, Example 2 Fig , p. 1131

35 STOKES’ THEOREM 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: Example 2

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

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 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. Equation 3

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 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.

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

42 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.

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

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

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 , 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.