2 cos sin 2 2. cos sin 2 cos 1 sin 2. .But = ( ),and = + = ( ) ( ) ( ) 2. [by the equality of mixed partials] = ± ( ) ( ).

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1 36 CHAPTER 6 VECTOR CALCULUS ( ) ( ) ( ) ( ) 3 6 ( ) 4 ( 3. Thus, as in the example, ) and F r F r. We parametrize as r() cos sin,. Then (cos )(sin ) sin cos F r F r cos sin sin cos cos sin cos 3 cos sin cos sin cos sin 9. Since is a simple closed path which doesn t pass through or enclose the origin, there exists an open region that doesn t contain the origin but does contain. Thus ( ) and ( ) have continuous partial derivatives on this open region containing and we can apply Green s Theorem. But by Exercise (a),,so F r. 3. Using the firstpartof(5),wehavethat ().But ( ),and, and we orient by taing the positive direction to be that which corresponds, under the mapping, to the positive direction along,so ( ) ( ) ( ) ± ± ± ( ) ( ) ( ) ( ) [using Green s Theorem in the -plane] [using the Chain Rule] [by the equality of mixed partials] ± () () The sign is chosen to be positive if the orientation that we gave to corresponds to the usual positive orientation, and it is negative otherwise. In either case, since () is positive, the sign chosen must be the same as the sign of Therefore () ( ) ( ). ( ) ( ). 6.5 Curl and Divergence. (a) curl F F ( ) ( ) i ( ) ( ) ( ) ( ) ( ) i ( ) ( ) c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

2 (b) div F F ( ) ( ) ( ) 3 3. (a) curl F F ( ) i ( ) ( ) ( ) (b) div F F ( ) () ( ) ( ) 5. (a) curl F F [( ) i ( ) ( ) ] ( ) 3 (b) div F F SECTION 6.5 CURL AND DIVERGENCE 37 ( ) 3 ( ) 3 ( ) 3 ( ) 3 7. (a) curl F F ( cos ) i ( cos ) ( cos ) sin sin sin h cos cos cos i (b) div F F ( sin ) ( sin ) ( sin ) sin sin sin 9. If the vector field is F, then we now. In addition, the -component of each vector of F is,so, hence. decreases as increases, so,but doesn t change in the -or-directions, so. (a) div F (b) curl F ( ) ( ) ( ). If the vector field is F, then we now. In addition, the -component of each vector of F is,so,hence. increases as increases, so the -or-directions, so. (a) div F,but doesn t change in c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

3 38 CHAPTER 6 VECTOR CALCULUS (b) curl F ( ) ( ) Since, is a vector pointing in the negative -direction. 3. curl F F (6 6 ) i (3 3 ) ( 3 3 ) and F is definedonallofr 3 with component functions which have continuous partial derivatives, so by Theorem 4, F is conservative. Thus, there exists a function such that F. Then ( ) 3 implies ( ) 3 ( ) and ( ) 3 ( ). But ( ) 3,so( ) () and ( ) 3 (). Thus ( ) 3 () but ( ) 3 so (), a constant. Hence a potential function for F is ( ) curl F F (6 6 ) i (6 6 ) (4 3 6 ) 6 ( ) ( 3) 6 so F is not conservative. 7. curl F F [ ( )] i ( ) ( ) F is definedonallofr 3, and the partial derivatives of the component functions are continuous, so F is conservative. Thus there exists a function such that F. Then ( ) implies ( ) ( ) ( ) ( ). But ( ),so( ) () and ( ) (). Thus ( ) () but ( ) so () and a potential function for F is ( ). 9. No. Assume there is such a G. Thendiv(curl G) ( sin ) (cos ) ( ) sin sin 6, which contradicts Theorem.. curl F ( ) ( ) ( ). Hence F () () () () () () is irrotational. c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

4 SECTION 6.5 CURL AND DIVERGENCE 39 For Exercises 3 9, let F( ) and G( ). 3. div(f G) divh i ( ) ( ) ( ) divh divh i divf divg 5. div(f) div( h i) divh i () () h i () div F F 7. div(f G) (F G) G curl F F curl G 9. curl(curl F) ( F) Now let s consider grad(div F) F and compare with the above. (Note that F is defined on page 9 [ET 95].) [continued] c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

5 3 CHAPTER 6 VECTOR CALCULUS grad(div F) F Then applying Clairaut s Theorem to reverse the order of differentiation in the second partial derivatives as needed and comparing, we have curl curl F graddivf F as desired. 3. (a) (b) r (c) () i ( ) 3 r 3 r () () () () () () () (d) ln ln( ) ln( ) () r 33. By (3), ( ) n div( ) [ div( ) ] by Exercise 5. But div( ). Hence ( ) n. 35. Let ( ).Then and Green s first identity (see Exercise 33) says ( ) n n. But is harmonic on,so n and n ( n). 37. (a) We now that, and from the diagram sin (sin) w r. Butv is perpendicular to both w and r,sothatv w r. c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

6 SECTION 6.6 PARAMETRIC SURFACES AND THEIR AREAS 3 (b) From (a), v w r ( ) ( ) ( ) (c) curl v v () () ( ) () () ( ) [ ( )] w 39. For any continuous function on R 3,defineavectorfield G( ) h( ) i where ( ) ( ). Then div G (( )) () () ( ) ( ) by the Fundamental Theorem of Calculus. Thus every continuous function on R 3 is the divergence of some vector field. 6.6 Parametric Surfaces and Their Areas. (7 4) lies on the parametric surface r( ) h 3 5 i if and only if there are values for and where 3 7, 5,and 4.Butsolvingthefirst two equations simultaneously gives, and these values do not satisfy the third equation, so does not lie on the surface. (5 5) lies on the surface if 3 5, 5,and 5for some values of and. Solving the first two equations simultaneously gives 4, and these values satisfy the third equation, so lies on the surface. 3. r( ) ( ) (3 ) (4 5) h 3 h 4 h 5i. From Example 3, we recognize this as a vector equation of a plane through the point ( 3 ) and containing vectors a h 4i and b h 5i. Ifwe wish to find a more conventional equation for the plane, a normal vector to the plane is a b 4 4i 5 andanequationoftheplaneis4( ) ( 3) ( ) or r( ), so the corresponding parametric equations for the surface are,,.forany point ( ) onthesurface,wehave. With no restrictions on the parameters, the surface is,which we recognize as a hyperbolic paraboloid. c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.