The Stokes Theorem. (Sect. 16.7) The curl of a vector field in space

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Cuthbert Dorsey
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1 The tokes Theorem. (ect. 6.7) The curl of a vector field i space. The curl of coservative fields. tokes Theorem i space. Idea of the proof of tokes Theorem. The curl of a vector field i space Defiitio The curl of a vector field F = F, F, F 3 i R 3 is the vector field curl F = ( F 3 3 F ) i + ( 3 F F 3 ) j + ( F F ) k. Remark: ice the followig formula holds, i j k curl F = 3 F F F 3 the oe also uses the otatio curl F = F. Remark: The curl of a vector field measures the rotatioal compoet of the vector field at ever poit of its domai. I 3-dimesios a vector is eeded to collect this iformatio.

2 The curl of a vector field i space Fid the curl of the vector field F =,,. olutio: ice curl F = F, we get, i j k F = = ( ( ) () ) i ( ( ) () ) j + ( () () ) k, We coclude that = ( ) i ( ) j + ( ) k, F = ( + ),,. The tokes Theorem. (ect. 6.7) The curl of a vector field i space. The curl of coservative fields. tokes Theorem i space. Idea of the proof of tokes Theorem.

3 The curl of coservative fields Recall: A vector field F : R 3 R 3 is coservative iff there eists a scalar field f : R 3 R such that F = f. Theorem If a vector field F is coservative, the F =. Remark: This Theorem is usuall writte as ( f ) =. The coverse is true ol o simple coected sets. That is, if a vector field F satisfies F = o a simple coected domai D, the there eists a scalar field f : D R 3 R such that F = f. Proof of the Theorem: F = ( f f ), ( f f ), ( f f ) The curl of coservative fields Is the vector field F =,, coservative? olutio: We have show that F = ( + ),,. ice F, the F is ot coservative. Is the vector field F = 3, 3, 3 coservative i R 3? olutio: Notice that F = i j k = (6 6 ), (3 3 ), ( 3 3 ) =. ice F = ad R 3 is simple coected, the F is coservative, that is, there eists f i R 3 such that F = f.

4 The tokes Theorem. (ect. 6.7) The curl of a vector field i space. The curl of coservative fields. tokes Theorem i space. Idea of the proof of tokes Theorem. tokes Theorem i space Theorem The circulatio of a differetiable vector field F : D R 3 R 3 aroud the boudar of the orieted surface D satisfies the equatio F dr = ( F) dσ, where dr poits couterclockwise whe the uit vector ormal to poits i the directio to the viewer (right-had rule). r (t) r (t)

5 tokes Theorem i space Verif tokes Theorem for the field F =,, o the ellipse = {(,, ) : 4 + 4, = }. olutio: We compute both sides i F dr = ( F) dσ. We start computig the circulatio itegral o the ellipse + =. If we choose the upward ormal to, we have to choose a couterclockwise parametriatio for. o we choose, for t [, π], r(t) = cos(t), si(t),. ad the right-had rule ormal to =,,. tokes Theorem i space Verif tokes Theorem for the field F =,, o the ellipse = {(,, ) : 4 + 4, = }. olutio: Recall: F dr = ( F) dσ, with r(t) = cos(t), si(t),, t [, π] ad =,,. The circulatio itegral is: = π F dr = π F(t) r (t) dt cos (t), cos(t), si(t), cos(t), dt. F dr = π [ cos (t) si(t) + 4 cos (t) ] dt.

6 tokes Theorem i space Verif tokes Theorem for the field F =,, o the ellipse = {(,, ) : 4 + 4, = }. π [ olutio: F dr = cos (t) si(t) + 4 cos (t) ] dt. The substitutio o the first term u = cos(t) ad du = si(t) dt, implies ice π π cos (t) si(t) dt = F dr = π 4 cos (t) dt = u du =. π cos(t) dt =, we coclude that [ + cos(t) ] dt. F dr = 4π. tokes Theorem i space Verif tokes Theorem for the field F =,, o the ellipse = {(,, ) : 4 + 4, = }. olutio: F dr = 4π ad =,,. We ow compute the right-had side i tokes Theorem. I = ( F) dσ. i j k F = F =,,. is the flat surface { +, = }, so dσ = d d.

7 tokes Theorem i space Verif tokes Theorem for the field F =,, o the ellipse = {(,, ) : 4 + 4, = }. olutio: F dr = 4π, =,,, F =,,, ad dσ = d d. The, ( F) dσ =,,,, d d. The right-had side above is twice the area of the ellipse. ice we kow that a ellipse /a + /b = has area πab, we obtai ( F) dσ = 4π. This verifies tokes Theorem. tokes Theorem i space Remark: tokes Theorem implies that for a smooth field F ad a two surfaces, havig the same boudar curve holds, ( F) dσ = ( F) dσ. Verif tokes Theorem for the field F =,, o a half-ellipsoid = {(,, ) : + + =, }. a olutio: (The previous eample was the case a.) a We must verif tokes Theorem o, F dr = ( F) dσ.

8 tokes Theorem i space Verif tokes Theorem for the field F =,, o a half-ellipsoid = {(,, ) : + + =, }. a olutio: F dr = 4π, F =,,, I = ( F) dσ. a is the level surface F = of F(,, ) = + + a. = F F, F =,, a, ( F) = /a F. dσ = F F k = F /a ( F) dσ =. tokes Theorem i space Verif tokes Theorem for the field F =,, o a half-ellipsoid = {(,, ) : + + =, }. a olutio: F dr = 4π ad ( F) dσ =. Therefore, ( F) dσ = d d = (π). We coclude that ( F) dσ = 4π, o matter what is the value of a >.

9 The tokes Theorem. (ect. 6.7) The curl of a vector field i space. The curl of coservative fields. tokes Theorem i space. Idea of the proof of tokes Theorem. Idea of the proof of tokes Theorem plit the surface ito surfaces i, for i =,,, as it is doe i the figure for = 9. F dr = = i= i= i i i= F dr i F d r i Ri ( F) dσ. ( i the border of small rectagles); ( F) i da (Gree s Theorem o a plae);

Stokes Theorem is an extension of Green s Theorem. It states that given a. (curl F) n ds

Stokes Theorem is an extension of Green s Theorem. It states that given a. (curl F) n ds 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