M 312 S T S P 1. Calculate the integral F dr where F = x + y + z,y + z, z and C is the intersection of the plane. x = y and the cylinder y 2 + z 2 =1

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1 M T P. alculate the itegral F dr where F = + +, +, ad is the itersectio of the plae = ad the clider + = (a) directl, (b) b tokes theorem.. Verif tokes theorem o the triagle with vertices (,, ), (,, ), (,, ) ad F = +,, +.. Verif tokes theorem for v =, +, + ad is the curve of itersectio of the sphere + + =5ad the plae = Verif tokes theorem for F =,, over the part of the paraboloid = ( + ) for which. 5. alculate the itegral d d + d with the itersectio of the plae + + =ad the clider + = (a) directl, (b) b tokes theorem.. Let be the part of the surface = with ad orieted upwards ad let F =,,. alculate curl F d (a) directl, (b) b tokes theorem. 7. Verif tokes theorem for F =,, over the part of the sphere + + =for which,. 8. Verif tokes theorem for F =,, over the part of the surface + = iside the first octat with. 9. Verif tokes theorem for F =,, over the part of the surface + =9iside the first octat with 4 ad.. osider the poits K(,, ), L(,, ), M(,, ), ad N(,, ). (a) how that K, L, M, ad N are coplaar. (b) Let be made up of lie segmets: from K to L, from L to M, from M to N, ad 4 from N to K. Evaluate d+ d + d directl. (c) Evaluate the lie itegral i part (b) b tokes theorem.

tokes Theorem upplemetar Problems - OLUTION KEY. alculate the itegral F dr where F = + +, +, ad is the itersectio of the plae = ad the clider + = (a) directl (b) b tokes theorem.

2 tokes Theorem upplemetar Problems - OLUTION KEY. alculate the itegral F dr where F = + +, +, ad is the itersectio of the plae = ad the clider + = (a) directl (b) b tokes theorem. Part (a): directl, The curve of itersectio of = ad + =ca be parameteried usig t goig from to π : =cost, =cost, =sit (we are takig a stadard parameteriatio aroud the circle + =, the set = ). Therefore, d = si tdt, d = si tdt, d =costdt. F dr = ( + + )d +( + )d + d = ( + )d + d (cotributios from d, d, ad d o a closed path are all ero). F dr = π [ (cos t +sit)( si t) si t ] dt = π ( si t cos t si t)dt = π ( si t cos t +cost)dt = [ si t t + si t] π = π. Part (b): b tokes theorem. The iterior of the curve from part (a) ca be projected oto the uit disc i the -plae. The origial surface has a equatio =. A ormal vector to the plae =is,,. Our uit ormal is = ±,,. hoose + for cosistec with the orietatio of chose i part (a). / / / ( ) = ( + ) i ( ( + + ))j +( ( + ) ( + + ))k =,, ( ) / {}}{ dd = (curl F ) d = R sice R is the uit disc i the -plae. dd = (area of R) = π R

3 . Verif tokes theorem o the triagle with vertices (,, ), (,, ), (,, ) ad F = +,, +. LH = F dr = ( + )d + d +( + )d = d + d sice cotributios from d, d, ad d o a closed path are all ero. = segmet from (,, ) to (,, ) ca be parameteried usig t : = t; =t; =;d = dt; d =dt; d =;t goes from to : d + d = (t) ( )dt = 8 [t ] = 8 segmet from (,, ) to (,, ) ca be parameteries usig t : =; = t; =t; d =;d = dt; d =dt; t goes from to : d + d = segmet from (,, ) to (,, ) ca be parameteried usig t : =t; =; = t; d =dt; d =;d = dt; t goes from to : d + d = (t) ( )dt = 8 [t ] = 8 F dr = =. RH = (curl F ) d / / / =()i ()j +( )k =,,. + + The plae cotaiig the three poits has a ormal vector =(4)i ( 4)j +(4)k = 4, 4, 4 or,,. The plae equatio is ( ) + + =, i.e. + + =. The uit ormal is ±.,, For cosistec with the orietatio of chose i part (a), we select the + sig. curl F = ( + ). Projectigotothe-plaeweobtai (curl F ) d = = [ + = ( 4 ) d = ] = = d = / {}}{ ( + ) dd = (( )+ ( ) [ ] 4 = 8 8= ) d ( + )dd =

. Verif tokes theorem for v =, +, + ad is the curve of itersectio of the sphere + + =5ad the plae = 4.

This curve ca be parameteried usig θ goig from to π : =cosθ; = si θ; = 4; d = siθdθ; d =cosθdθ; d =; v dr = ( )d +( + )d +( + )d = π ( 7 si θ 8 si θ cos θ +7cos θ +54cos θ ) dθ

4 . Verif tokes theorem for v =, +, + ad is the curve of itersectio of the sphere + + =5ad the plae = LH = v dr The sphere itersects the plae alog the circle with radius cetered at (,, 4). This curve ca be parameteried usig θ goig from to π : =cosθ; = si θ; = 4; d = siθdθ; d =cosθdθ; d =; v dr = ( )d +( + )d +( + )d = π ( 7 si θ 8 si θ cos θ +7cos θ +54cos θ ) dθ = 7π RH = (curl v ) d curl v = / / / + + =()i ( + )j +(+4 + )k =,, +4 + Takig to be the disc = 4, + 9, the uit ormal to is = k (orietatio chose to be cosistet with the orietatio of ) curl v k =+4 + = sice = 4. Usig clidrical coordiates, (curl v ) d = π ( r si θ)rdrdθ= [ ] π r r= r si θ dθ r= = π ( 7 8 si θ) dθ = [ 7 θ +8cosθ] π = 7π

5 4. Verif tokes theorem for F =,, over the part of the paraboloid = ( + ) for which. LH = F dr ubstitutig = ito the paraboloid equatio ields =( + ), i.e., + = 4. The curve is a circle that ca be parameteried usig θ goig from to π : ; d = cos θdθ; d = = cos θ; = si θ; = F dr = d d + d = π si θdθ; d = ( 4 si θ 4 cos θ ) dθ = 4 π dθ = 4 [θ]π = π RH = (curl F ) d / / / =() i ()j +( )k =,,. The paraboloid ca be cosidered a level surface of g(,, ) = +. = ± g g = ± 4,4, - we must pick the sig for cosistec with the orietatio of curl F = ( 4 ) Project the surface oto the disc D : + 4 i the -plae: / (curl F ) d = D ( 4 ( ( + )) {}}{ ) + +d d + + witch to polar coordiates: π / ( 8r cos θ )rdrdθ= [ ] π r=/ 8r 5 cos θ r dθ 5 r= = π ( cos θ ) 4 dθ = [si θ]π 4 [θ]π = π.

5. alculate the itegral d d + d with the itersectio of the plae + + = ad the clider + = (a) directl (b) b tokes Theorem ++= += Part (a): directl Let us begi b completig the square i the clider s

The itersectio of the clider ad a plae is a ellipse that ca be parameteried usig t goig from to π : =cost =+sit = cost si t (obtaied from the plae equatio, = ) Therefore, d = si tdt; d =costdt; d

6 5. alculate the itegral d d + d with the itersectio of the plae + + = ad the clider + = (a) directl (b) b tokes Theorem ++= += Part (a): directl Let us begi b completig the square i the clider s equatio: + = + = + + = +( ) = The clider s geeratig curve is a circle i the -plae cetered at (,, ), radius. The itersectio of the clider ad a plae is a ellipse that ca be parameteried usig t goig from to π : =cost =+sit = cost si t (obtaied from the plae equatio, = ) Therefore, d = si tdt; d =costdt; d =(sit cos t) dt LH = F dr = d d + d = π [( + si t)( si t) ( cost si t)cost + ( + si t)( si t cos t)] dt = π [ sit si t +cos t +cost +sitcos t +sit + si t cos t si t cos t ] dt = π cos tdt = π ( + cos t) dt = [ θ + si θ] π = π Part (b): b tokes theorem. RH = (curl F ) d / / / =()i ()j +( )k =,,. A ormal vector to the plae + + =is,,. A uit ormal is = ±,,. hoose + for cosistec with the orietatio of chose i part (a). curl F = Project oto the circular disc D : +( ) i the -plae: / (curl F ) d = d = {}}{ D d d = d d =(area of D) = π D

7 . Let be the part of the surface = with ad orieted upwards ad let F =,,. alculate curl F d (a) directl, 4 (b) b tokes theorem. Part (a): directl / / / =()i ()j +()k =,, is a level surface of g(,, ) = (with g =);auitormalvectorto is ± g g = ±,, choose + for upward orietatio: =,, ; curl F = + Project oto the -plae so that curl F d = curl F d = dd= [ ] [] = / {}}{ ( ) + d d =dd + ; Part (b): b tokes theorem LH = F dr = d + d + d; = 4 segmet from (,, ) to (,, ) ca be parameteried usig goig from to : =; =;d =;d =; d + d + d = d =[] = ca be parameteried usig goig from to : =; =/; d =;d = / d; d + d + d = ( ) 9 d =[9 +/] = 5 segmet from (,, ) to (,, ) ca be parameteried usig goig from to : =; = ; d =;d =; d + d + d = d = [] = 4 ca be parameteried usig goig from to : =; =/; d =;d = / d; 4 d + d + d = d = F dr =+5 += 9.

8 7. Verif tokes theorem for F =,, over the part of the sphere + + =for which,. LH = F dr = d d + d; = ca be parameteried usig θ =cosθ, =siθ, =,d= si θdθ,d=cosθdθ, d=;θ goes from π to π ; d d + d = π/ ( π/ si θ cos θ ) dθ = π/ dθ =[ θ]π/ π/ π/ = π has =ad d =therefore d d + d = F dr = π += π RH = (curl F ) d / / / =() i ()j +( )k =,, is a level surface of g(,, ) = + + (with g =); auitormalvectorto is ± g g = ±,, = ±,, ; choose + for orietatio to agree with the curve : =,, ; curl F = Itegrate usig spherical coordiates with ρ =, d = si φdφdθ (curl F ) d = π/ π/ ( π/ (cos θ si φ) (si θ si φ) cosφ ) si φdφdθ = π/ π/ π/ cos(θ)dθ si φdφ π/ π/ dθ π/ cos φ si φdφ }{{} = [ ] π/ si(θ) π/ substitute u=cos φ ] π/ [ cos φ cos φ [θ] π/ π/ [ ] si π/ φ = π

9 8. Verif tokes theorem for F =,, over the part of the surface + = iside the first octat with. 4 LH = F dr = ( ) d; = 4 ca be parameteried usig θ goig from to π/ =cosθ, =siθ, =;d = si θdθ,d=cosθdθ, d= ( ) d = ca be parameteried usig goig from to =,= ; d =,d= d ( ) d = [ ] d = = 7 ca be parameteried usig θ goig from π/ to =cosθ, =siθ, =;d = si θdθ,d=cosθdθ, d= ( ) d = 4 ca be parameteried usig goig from to =,= ; d =,d= d ( 4 ) d = [ ] d = F dr = = 4 = 7 RH = (curl F ) d / / / =( ) i ()j +()k =,, is a level surface of g(,, ) = + (with g =); auitormalvectorto is ± g g = 4 ±,, ; choose + for orietatio to agree with the curve : = curl F = 8 4 = ρ [ ] π/ [ ] = si θ ρ,, ; Itegrate usig spherical coordiates with φ = π 4, d = ρ si π 4 dρ dθ = ρ dρ dθ; = ρ si π 4 cos θ = ρ cos θ; = ρ si π 4 si θ = ρ si θ (curl F ) d = ( )( ) π/ 4 ρ cos θ ρ si θ ρ ρ dρ dθ = 4 π/ cos θ si θdθ ( = ) ( ) = 4 ρ dρ

10 9. Verif tokes theorem for F =,, over the part of the surface + =9iside the first octat with 4 ad. 4 4 LH = F dr = ( ) d +( )d +( ) d = d d d sice cotributios from d, d, ad d o a closed path are all ero. = 4 ca be parameteried usig θ goig from to π 4 =cosθ, = si θ, =;d = siθdθ, d=cosθdθ, d = d d d = π/4 9si θdθ = 9 π/4 ( cos(θ)) dθ = [ 9θ 9 4 si(θ)] π/4 = 9π ca be parameteried usig goig from to 4 =,=,d=,d= d d d = 4 d = ca be parameteried usig θ goig from π 4 to =cosθ, = si θ, =4;d = siθdθ, d=cosθdθ, d = d d d = ( π/4 9si θ cos θ ) dθ = [ 9θ 9 4 si(θ) si θ] = 9π π/ ca be parameteried usig goig from 4 to =,=,d=,d= 4 d d d = d = 4 F dr = 9π π = RH = (curl F ) d / / / =()i ( )j +()k =,, is a level surface of g(,, ) = + (with g =9); auitormalvectorto is ± g g = ±,, = ± 4 +4,, ; choose + for orietatio to agree with the curve : =,, ; curl F = + Use clidrical coordiates with r =,d =d dθ (curl F ) d = π/4 4 cosθ+ si θ d dθ = [si θ cos θ] π/4 [] 4 =( +)(4) =

4 N M K L Part (a): begi b formig a equatio of the plae passig through K, L, ad M : KL KM = =4i ( 4)j +()k = 4, 4, ields a plae equatio 4( ) + 4 + =, i.e., + + =4.

11 . osider the poits K(,, ), L(,, ), M(,, ), ad N(,, ). (a) how that K, L, M, ad N are coplaar. (b) Let be made up of lie segmets: from K to L, from L to M, from M to N, ad 4 from N to K. Evaluate d+ d + d directl. (c) Evaluate the lie itegral i part (b) b tokes theorem. 4 N M K L Part (a): begi b formig a equatio of the plae passig through K, L, ad M : KL KM = =4i ( 4)j +()k = 4, 4, ields a plae equatio 4( ) =, i.e., + + =4. ice N(,, ) satisfies this equatio, the four poits are coplaar. Part (b) F dr = d + d + d; = 4 ca be parameteried usig goig from to : =, =;d = d, d = d + d + d = [ ] ( )d = = ca be parameteried usig goig from to : =,=4, d =,d= d d + d + d = d = [ ] = ca be parameteried usig goig from to : =, =;d = d, d = d + d + d = [ ] ( )d = = 4 ca be parameteried usig goig from to : =,=4, d =,d= d 4 d + d + d = F dr = ++ += Part (c): (curl F ) d / / / =()i ()j +( )k =,, From part (a), =,, (chose the orietatio to agree with ); curl F = + Projectigotothe-plae (curl F ) d = / + / {( }}{ ) d d = [ + ] = / = d = = +

v n ds where v = x z 2, 0,xz+1 and S is the surface that

v n ds where v = x z 2, 0,xz+1 and S is the surface that M D T P. erif the divergence theorem for d where is the surface of the sphere + + = a.. Calculate the surface integral encloses the solid region + +,. (a directl, (b b the divergence theorem. v n d where