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

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1 VECTOR CALCULU 16.8 tokes Theorem In this section, we will learn about: The tokes Theorem and using it to evaluate integrals.

2 TOKE V. GREEN THEOREM tokes 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. tokes Theorem relates a surface integral over a surface to a line integral around the boundary curve of (a space curve).

3 INTRODUCTION Oriented surface with unit normal vector n. The orientation of induces the positive orientation of the boundary curve C. 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.

4 TOKE THEOREM Let: 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 R 3 that contains. Then, C F dr = curl F d

5 TOKE THEOREM The theorem is named after the Irish mathematical physicist ir George tokes ( ). What we call tokes Theorem was actually discovered by the cottish physicist ir William Thomson ( , known as Lord Kelvin). tokes learned of it in a letter from Thomson in 1850.

6 TOKE THEOREM C and F dr = FT ds C curl F d= curl Fn d Thus, tokes Theorem says: The line integral around the boundary curve of of the tangential component of F is equal to the surface integral of the normal component of the curl of F.

7 TOKE THEOREM Equation 1 The positively oriented boundary curve of the oriented surface is often written as. o, the theorem can be expressed as: curl F d= F dr

8 TOKE THEOREM, GREEN THEOREM, & FTC There is an analogy among tokes 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.

9 TOKE THEOREM, GREEN THEOREM, & FTC In fact, consider the special case where the surface is flat, in the xy-plane with upward orientation. Then: The unit normal is k. The surface integral becomes a double integral. tokes Theorem becomes: C F dr = curl F d= curl F kda ( ) Thus, we see that Green s Theorem is really a special case of tokes Theorem.

10 TOKE THEOREM Evaluate F d r, 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.) d r could be evaluated F C C directly, however, it s easier to use tokes Theorem. Example 1

11 TOKE THEOREM Example 1 We first compute for F(x, y, z) = y 2 i + x j + z 2 k: i j k curl F= = 1+ 2y x y z y x z 2 2 ( ) k

12 TOKE THEOREM Example 1 There are many surfaces with boundary C. The most convenient choice, though, is the elliptical region in the plane y + z = 2 that is bounded by C. If we orient upward, C has the induced positive orientation.

13 TOKE THEOREM Example 1 The projection D of on the xy-plane is the disk x 2 + y 2 1. o, using Equation 10 in ection 16.7 with z = g(x, y) = 2 y, we have the following result.

14 TOKE THEOREM Example 1 C F dr = curl F d= 1+ 2y da 1 2 D ( ) 2π 1 = 1+ 2rsinθ 0 0 2π 0 ( ) 2 3 2π r r = 2 sinθ = + ( 1 2sinθ) ( ) 2 3 = 2π + 0= π dθ rdrdθ 1 0 dθ

15 TOKE THEOREM Example 2 Use tokes Theorem to compute curl F d where: F(x, y, z) = xz i + yz j + xy k 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.

16 TOKE THEOREM Example 2 To find the boundary curve C, we solve: x 2 + y 2 + z 2 = 4 and x 2 + y 2 = 1 ubtracting, we get z 2 = 3, and (since z > 0), z = z = 3 o, C is the circle given by: x 2 + y 2 = 1, 3

17 TOKE THEOREM Example 2 A vector equation of C is: r(t) = cos t i + sin t j + 3 k 0 t 2π Therefore, r (t) = sin t i + cos t j Also, we have: ( ( )) = 3 cos + 3 sin + cos sin F r t ti tj t tk

18 TOKE THEOREM Thus, by tokes Theorem, Example 2 curl F d= F dr C 2π = Fr ( ()) t r'() t dt 0 2π 0 ( 3 cos sin 3 sin cos ) = t t + t t dt 2π = 3 0dt = 0 0

19 TOKE 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! In general, if 1 and 2 are oriented surfaces with the same oriented boundary curve C and both satisfy the hypotheses of tokes Theorem, then curl F d= F dr = curl F d 1 2 C This fact is useful when it is difficult to integrate over one surface but easy to integrate over the other.

20 CURL VECTOR We now use tokes Theorem to throw some light on the meaning of the curl vector. uppose that C is an oriented closed curve and v represents the velocity field in fluid flow. 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. C v dr = vt ds C

21 CIRCULATION Thus, v d r C is a measure of the tendency of the fluid to move around C. It is called the circulation of v around C.

22 CURL VECTOR Now, let: P 0 (x 0, y 0, z 0 ) be a point in the fluid, and 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 a because curl F is continuous. Thus, by tokes Thm., we get the following approximation to the circulation around the boundary circle C a : v dr = curl v d= curl v nd C a a curl v ( P ) n( ) 0 0 ( ) n( ) = curl v P P π a 0 0 The approximation becomes better as a 0. Thus, we have: 1 curl v( P0) n( P0) = lim v dr a 0 2 π a C a a a P d 2

23 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. 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.

24 VECTOR CALCULU 16.9 The Divergence Theorem In this section, we will learn about: The Divergence Theorem for simple solid regions, and its applications in electric fields and fluid flow.

25 INTRODUCTION In ection 16.5, we rewrote Green s Theorem in a vector version as: C Fn ds = D div F( x, y) da, where C is the positively oriented boundary curve of the plane region D. If we were seeking to extend this theorem to vector fields on R 3, we might make the guess that Fn d = E div F( x, y, z) dv where is the boundary surface of the solid region E. It turns out that this is true, under appropriate hypotheses, and is called the Divergence Theorem.

26 IMPLE OLID REGION We state the Divergence Theorem for regions E that are simultaneously of types 1, 2, and 3. We call such regions simple solid regions. For instance, regions bounded by ellipsoids or rectangular boxes are simple solid regions. The boundary of E is a closed surface. We use the convention, introduced in ection 16.7, that the positive orientation is outward. That is, the unit normal vector n is directed outward from E.

27 THE DIVERGENCE THEOREM Let: E be a simple solid region and let be the boundary surface of E, given with positive (outward) orientation. F be a vector field whose component functions have continuous partial derivatives on an open region that contains E. Then, F d= div FdV E

28 THE DIVERGENCE THEOREM Thus, the Divergence Theorem states that: Under some conditions, the flux of F across the boundary surface of E is equal to the triple integral of the divergence of F over E. The Divergence Theorem is sometimes called Gauss Theorem after the great German mathematician Karl Friedrich Gauss ( ) (discovered during his investigation of electrostatics). In Eastern Europe, it is known as Ostrogradsky s Theorem (published in 1826) after the Russian mathematician Mikhail Ostrogradsky ( ).

29 DIVERGENCE Example 1 Find the flux of the vector field F(x, y, z) = z i + y j + x k over the unit sphere x 2 + y 2 + z 2 = 1 First, we compute the divergence of F: div F = ( z) + ( y) + ( x) = 1 x y z

30 DIVERGENCE Example 1 The unit sphere is the boundary of the unit ball B given by: x 2 + y 2 + z 2 1 o, the Divergence Theorem gives the flux as: F d = div F dv = 1dV B B ( ) 4 π ( 1) 3 = V B = = 3 4π 3

31 DIVERGENCE Evaluate Example 2 F d where: F(x, y, z) = xy i + (y 2 + e xz2 ) j + sin(xy) k is the surface of the region E bounded by the parabolic cylinder z = 1 x 2 and the planes z = 0, y = 0, y + z = 2

32 DIVERGENCE Example 2 It would be extremely difficult to evaluate the given surface integral directly. We would have to evaluate four surface integrals corresponding to the four pieces of. Also, the divergence of F is much less complicated than F itself: divf 2 = xy + y + e + sin xy x y z = y+ 2y = 3y ( ) ( 2 xz ) ( )

33 DIVERGENCE Example 2 o, we use the Divergence Theorem to transform the given surface integral into a triple integral. The easiest way to evaluate the triple integral is to express E as a type 3 region: E = {( xyz,, ) 1 x 1,0 z 1 x 2,0 y 2 z}

34 DIVERGENCE Then, we have: Example 2 = = = F d 3 E E div F dv 3y dv x 2 z y dy dz dx

35 DIVERGENCE Example 2 = x = 2 ( ) ( ) ( 3 3 7) x 1 ( 2 ) 3 1 = 2 x dx 1 z z dz dx dx = x + x + x dx =

36 UNION OF IMPLE OLID REGION The Divergence Theorem can also be proved for regions that are finite unions of simple solid regions. For example, let s consider the region E that lies between the closed surfaces 1 and 2, where 1 lies inside 2. Let n 1 and n 2 be outward normals of 1 and 2. boundary surface of E is: = 1 2 Its normal n is given by: n = n 1 on 1 n = n 2 on 2

37 UNION OF IMPLE OLID RGN. Equation 7 Applying the Divergence Theorem to, we get: E div FdV = F d = Fn d ( ) = F n d + Fn = F d+ F d d Let s apply this to the electric field (Exp. 5 in ec.16.1): εq ( ) = 3 E x x x where 1 is a small sphere with radius a and center the origin.

38 APPLICATION ELECTRIC FIELD You can verify that div E = 0 (Exercise 23). Thus, the eq. from the previous slide gives: E d= E d+ divedv E 2 1 = E d 1 = En d 2 The point of this calculation is that we can compute the surface integral over 1 because 1 is a sphere.

39 APPLICATION ELECTRIC FIELD The normal vector at x is x/ x. Therefore. εq x εq εq εq En = x 3 = = 4 2 = 2 xx x x x x a since the equation of 1 is x = a.

40 APPLICATION ELECTRIC FIELD Thus, we have: εq E d= En d = d a = = = εq 2 A a εq 4 π 2 a a 4πεQ ( ) 1 2

41 APPLICATION ELECTRIC FIELD This shows that the electric flux of E is 4πεQ through any closed surface 2 that contains the origin. This is a special case of Gauss s Law (Equation 11 in ection 16.7) for a single charge. The relationship between ε and ε 0 is ε = 1/4πε 0.

42 APPLICATION FLUID FLOW Another application of the Divergence Theorem occurs in fluid flow. Let v(x, y, z) be the velocity field of a fluid with constant density ρ. Then, F = ρv is the rate of flow per unit area. uppose: P 0 (x 0, y 0, z 0 ) is a point in the fluid. B a is a ball with center P 0 and very small radius a. Then, div F(P) div F(P 0 ) for all points in B a since div F is continuous.

43 APPLICATION FLUID FLOW We approximate the flux over the boundary sphere a as follows: a F d= div FdV = B B a a = div F div F ( ) 0 ( P ) V ( B ) 0 P dv a This approximation becomes better as a 0 and suggests that: 1 divf( P ) 0 = lim F d a 0 V B ( ) a a

44 OURCE AND INK Equation 8 says that div F(P 0 ) is the net rate of outward flux per unit volume at P 0. (This is the reason for the name divergence.) If div F(P) > 0, the net flow is outward near P and P is called a source. If div F(P) < 0, the net flow is inward near P and P is called a sink.

45 OURCE For this vector field, it appears that the vectors that end near P 1 are shorter than the vectors that start near P 1. Thus, the net flow is outward near P 1. o, div F(P 1 ) > 0 and P 1 is a source.

46 INK Near P 2, the incoming arrows are longer than the outgoing arrows. The net flow is inward. o, div F(P 2 ) < 0 and P 2 is a sink.

47 OURCE AND INK We can use the formula for F to confirm this impression. ince F = x 2 i + y 2 j, we have div F = 2x + 2y, which is positive when y > x. o, the points above the line y = x are sources and those below are sinks.