III.g tokes Theorem tokes Theorem is a extesio of Gree s Theorem. It states that give a surface, i space, with its boudar, the F dr = (curl F) d where is a uit ormal to which alwas poits i the same directio relative to ad whose directio is related to the directio of travel alog b the right had rule: if the figers of the right had poit alog the directio of travel alog the there is a poit P of such that the thumb poits i the directio of at P. This statemet, as well as the techical mathematical coditios eeded o, ad a proof of the Theorem, ca be foud i precise form i theoretical texts. We merel poit out that the theorem is valid for, ecoutered i practice. The followig should be kept i mid. uppose we are give first a curve i space. The we ma costruct ma surfaces 1, 2,... whose boudar is. Thik of beig the rim of a soap bubble ad we blow o the soap film. 235
3 2 1 1 As aother example, ote that the uit circle x 2 + 2 = 1 ca be viewed as the boudar of the ellipsoids: x 2 + 2 + 2 /a 2 = 1,, for a a. I such cases, the theorem holds for the give ad a such. We pass to examples. Example 1. Let be the hemisphere = 1 x 2 2 ad suppose F = i xj + k. If is the upward poitig ormal to, verif tokes Theorem. Aswer. We eed to show (curl F) d = F dr with traversed as show. x It is usuall easier to evaluate F dr so we start with this. Now x = cos θ = = si θ = 236 θ 2π.
o F dr = = 2π 2π [ dx dθ xd dθ + d ] dθ dθ (cos θ)(cos θ) dθ = 1 2 2π [1 cos 2θ] dθ = 1 2 2π = π. We pass to the evaluatio of (curl F) d, ad calculate the pieces eeded first. Observe i j k curl F = det / x / / = i[1] j[ 1] + k[ 1] = i + j k. x Now we proceed b parametriig. We have a choice: we ca either use = 1 x2 2 ad x, as parameters, or use spherical coordiates. The former is easier to set up, but usuall the itegrals are harder, so we choose the latter optio. Note that oce agai: = cos ϕ x = si ϕ cos θ = si ϕ si θ θ 2π ϕ π 2. o r = si ϕ cos θ i + si ϕ si θ j + cos ϕ k ad d = r θ r ϕ dθ dϕ with a ormal N give b r θ r ϕ. We compute r θ r ϕ ad get i j k r θ r ϕ = det si ϕ si θ si ϕ cos θ cos ϕ cos θ cos ϕ si θ si ϕ = (si 2 ϕ cos θ i + si 2 ϕ si θ j + cos ϕ si ϕ k). 237
The ext questio is if N = r θ r ϕ poits the right wa (i.e., upwards) or ot. hoose a coveiet poit sa ϕ = π/4, a θ ad look at the k-compoet of N. We fid it is egative, so we must choose = N N = r θ r ϕ r θ r ϕ. o d = (r θ r ϕ )dθ dϕ. (Agai: There is o eed to actuall calculate r θ r ϕ!) We the have: (curl F) d = B (i + j k) (si 2 ϕ cos θ i + si 2 ϕ si θ j + cos ϕ si ϕ k)dϕ dθ where B deotes the rectagle θ 2π, ϕ π/2 i the (ϕ, θ) plae. 2π θ B π 2 φ osequetl, (curl F) d = as expected. = = 2π π/2 θ= ϕ= 2π π/2 θ= 2π ϕ= [si 2 ϕ cos θ + si 2 ϕ si θ cos ϕ si ϕ] dϕ dθ [ (cos θ + si θ) { [cos ] θ + si θ 2 [ ] 1 cos(2ϕ) 2 π } 2 dθ = π π si2 ϕ 2 ] cos ϕ si ϕ dϕ dθ Example 2. Verif tokes Theorem for the same F as i example 1, except ow is the upper ellipsoid: = a 1 x 2 2 (a > ) with the ormal poitig 238
upward. Aswer. Note that i this case ad the directio of travel are the same as i example 1. x It follows that F dr = π/2 regardless of the value of a! We must show that this is also true of (curl F) d ad sice we worked out curl F i example 1, we eed ol work out, d. Note that ow is the upper half of ( 2 /a 2 )+x 2 + 2 = 1. We use the same parametriatio as before, except is replaced b /a, i.e., we have: = a cos ϕ = si ϕ si θ x = si ϕ cos θ ϕ π 2 θ 2π. We repeat the earlier calculatios ad fid d = (r θ r ϕ ) dθ dϕ = [a si 2 ϕ cos θ i + a si 2 ϕ si θ j + cos ϕ si ϕ k] dθ dϕ ad ( F) d = 2π π/2 (a si 2 ϕ cos θ + a si 2 ϕ si θ cos ϕ si ϕ) dϕ dθ = π 239
oce agai! Example 3. Evaluate F dr b meas of tokes Theorem if is the itersectio of the plae = + 2 ad the clider x 2 + 2 = 1 traversed couterclockwise as viewed from above while F = (x 1 + )i + (ta 1 + x)j + (cos( 2 ) + x)k. Aswer. This problem meas that we are to evaluate F dr b actuall calcu- latig curl F d. Observe first that is ot give. We ca choose a that has for its boudar, so keep it simple! o we start with a sketch. It seems reasoable to use the give b the piece of the plae = + 2 iside x 2 + 2 = 1. x o, = + 2 = x = x x 2 + 2 1 = or x = r cos θ = = r si θ = r si θ + 2 24 r 1 θ 2π
is also fie. We the have d = r x r dx d ad N = r x r. Now r x r = det i j k 1 1 1 = j[1] + k[1]. We check if this N poits the right wa: Note that as we travel i the give directio alog the b the right had rule should poit upwards. ice this is true of N (b lookig at the k compoet), we coclude it poits the right wa ad d = N dx d = [ j + k] dx d. Next, curl F = det ad curl F d = 1 dx d, so i j k / x / / x 1 + ta 1 + x cos( 2 ) + x (curl F) d = sice B is the uit circle i the (x, ) plae. B = i() j(1) + k() = j dx d = π, Example 4. Evaluate b tokes Theorem (curl F) d if: is the part of the paraboloid x = 2 + 2 to the left of x = 1; poits towards decreasig x; F = i xj + k. 241
Aswer. I this case we eed to evaluate F dr where is the boudar of traversed i the correct directio. θ x Now is kow, ad from the picture, we coclude that must be traversed clockwise (as show) lookig towards decreasig x. Next o we have x = 1, 2 + 2 = 1 so we parametrie x = 1 = cos θ θ 2π, = si θ but this is, sice we traverse the wrog wa! The F dr = 2π [(cos θ) + ( 1)( si θ) + si θ cos θ]dθ =. o (curl F) d = F dr = =. To practice, let us also calculate (curl F) d ad see what happes. ice here F, are simple this ma be possible. Now i j k curl F = det / x / / = k[ 2] x 242
while is give b x = 2 + 2 = 2 + 2 1 = ad r = ( 2 + 2 )i + j + k. o r = 2i + j, r = 2i + k ad r r = det i j k 2 1 2 1 = i 2j 2k. Note that this poits the wrog wa (look at the i compoet). Thus (curl F) d = (curl F) ( r r )d d = ( 2)(2)d d = 4d d ad (curl F) d = B ( 4) d d = 2π 1 4(si θ)r dr dθ =. 2+ 2 = 1 B 243
Further Exercises: Verif tokes Theorem. 1) is the piece of plae x + + = 1 i the first octat, poits dowward, ad F = i + j + xk. 2) is the piece of the sphere x 2 + 2 + 2 = 1 iside the coe = 1 x 2 2, poits upward ad F = i + xj + k. 3) is the disc give b = 1, x 2 + 2 1, poits upward ad F = i + xj + ( + x)k. 4) is the piece of the coe = x 2 + 2 below = 1, poits upward ad F is as i Problem 3. Use tokes Theorem to evaluate. 5) F dr if is the itersectio of x2 + 2 + 2 = a 2 ad x 2 + 2 = ax with, traversed couterclockwise if lookig dowward, ad F = i + xj + ( + x)k. 6) F dr if is the boudar of the triagle with vertices (,, ), (1,, ), (, 1, 1) traversed couterclockwise if lookig dowward, ad F = x i + j + x k. 7) F dr if is obtaied b itersectig the plae = with the clider x 2 + 2 = 4, traversed clockwise if lookig towards icreasig, ad F = x i + ( x2 x2 + + x)j + ( 2 2 + 2 )k. 8) ( F) d if is the disc: x = 1, 2 + 2 1; poits towards icreasig x ad F = (x 3 si x)i + j k. 9) ame as Problem 8 if is the part of the paraboloid = x 2 + 2 iside the sphere x 2 + 2 + 2 = 2; poits dowward ad F = (si 2 )i+(cos 2 )j+x 2 k. 244
1) ame as Problem 8 if is the part of = x 2 + 2 which lies above x 1, 1, poits dowward ad F = x i + x 2 j + k. 245