10.9 Stokes's theorem

Transcription

1 09 tokes's theorem This theorem transforms surface integrals into line integrals and conversely, line integrals into surface integrals Hence, it generalizes Green's theorem in the plane of ec 04 Equation (99) will be used in ec 09, i j k curl v = v = (99) x y z v v v Theorem : tokes's theorem Let be a piecewise smooth oriented surface in space and let the boundary of be a piecewise smooth simple closed curve Let F ( xyz,, ) be a continuous vector function that has continuous first partial derivatives in a domain in space containing Then, (curl F) nda Fr( s) ds (09) Here n is a unit normal vector of and, depending on n, the integration around is taken in the sense shown in Fig 54 Furthermore, r() s dr / ds is the unit tangent vector and s is the arc length of In component form eqn (09) becomes, F F F F F F y z z x x y N N N du dv Fdx Fdy F dz (09) Here, F= Fi+ F j+ Fk, N= Ni+ N j+ Nk, nda = Ndu dv, r ds = dx i+ dy j+ dz k, and is the region with boundary curve in the uv plane corresponding to represented by r ( uv, ) The proof follows after Example June 0, /6

2 Example : Verification of tokes's theorem Verify tokes's theorem for F= yi+ zj+ xk and the paraboloid (see Fig 55) given by, z f xy xy z (, ) ( ), 0 olution: The curve, oriented as in Fig 55, is the circle r( s) cos sisin sj0k Its unit tangent vector is r ( s) = sin si+ cos sj The function F= yi+ zj+ xk on is Fr ( ( s)) sin si0 jcos sk Hence, π π (()) () sin π 0 0 Frdr Frs r s ds sds Now consider the surface integral ince F = yf, = zf, = x, we have curl F= i j k A normal vector of is N= grad( z f ( x, y)) = x i+ y j+ k Hence, curl F Nxy From eqn (06) nda = Ndu dv = N dx dy (with xy, instead of uv), Using polar coordinates r, θ defined by xrcos θ, y rsinθ and denoting the projection of into the xy plane by, (curl F) nda curl F N dx dy x y dx dy π θ 0 r0 θ θ rdrdθ r(cos sin ) π (cosθ sin θ) QED 0 d θ π θ June 0, /6

3 Proof: Proof of tokes s theorem Obviously, eqn (09) holds if the integrals of each component on both sides of eqn (09) are equal, ie, F F N N du dv Fdx z y, (09) F F N N du dv Fdx z x, (094) F F N N du dv Fdx y x (095) Prove this first for a surface that can be represented simultaneously in the forms, z f( xy, ), y gxz (, ), x hyz (, ) (096) Eqn (09) will be proved by using eqn (096a) etting u xv, y, from eqn (096a), r( uv, ) r( xy, ) xiyj f( xy, ) k From eqn (06) in ec 06, by direct calculation, Nr r r r f i f j k u v x y x y Note that N is an upper normal vector of, since it has a positive z component Also, the projection of into the xyplane,, with boundary curve (Fig 56) Hence, the left side of eqn (09) is, F F ( f y ) dx dy z y (097) Now consider the right side of eqn (09) Transform this line integral over into a double integral over by applying Green s theorem (eqn (04) of ec 04 with F 0 ), ie, June 0, /6

4 F F dx dx dy y Here, F (,, (, )) F xyf xy Hence, by the chain rule, F F F f ( xyf,, ( xy, )) ( xyz,, ) ( xyz,, ) y y z y where z f( xy, ) The right side of the above eqn equals the integrand in eqn (097) This proves eqn (09) Equations (094) and (095) follow in the same way if eqns (096b) and (096c) are used, respectively By addition eqn (09) is obtained This proves tokes s theorem for a surface that can be represented simultaneously in the forms given by eqn (096 a-c) As in the proof of the divergence theorem, the result may be immediately extended to a surface that can be decomposed into finitely many pieces, each of which is of the kind just considered This covers most of the cases of practical interest The proof in the case of a most general surface satisfying he assumptions of the theorem would require a limit process; this is similar to the situation in the case of Green s theorem in ec 04 Example : Green s theorem in the plane as a special case of tokes's theorem Let F= Fi+ F j be a vector function that is continuously differentiable in a domain in the xy, plane containing a simply connected bounded closed region whose boundary is a piecewise smooth simple closed curve Then according to eqn (09), F F (curl F) n(curl F) k x y Hence the formula in tokes s theorem now takes the form, x y F F da F dx F dy ( ) This shows that Green s theorem in the plane (ec 04) is a special case of tokes s theorem (which was needed in the proof of the latter) June 0, /6

5 Example : Evaluation of a line integral by tokes's theorem Evaluate F r () s ds, where is the circle x y z 4,, oriented counterclockwise as seen by a person standing at the origin, and with respect to righthanded artesian coordinates, F= y i+ xz j zy k olution: As a surface bounded by, the plane circular disk x y 4 in the plane z, can be taken Then n in the tokes s theorem points in the positive z direction; thus n k Hence (curl F) n is simply the component of curlf in the positive z direction With z, F= yi 7xj+ y k, F F (curl F) n = = 8 x y Hence, (curl F) nda 8 (area 4 π of the disk ) π an confirm this by direct calculation Example 4: Physical meaning of the curl in fluid motion irculation Let be a circular disk of radius and center P bounded by the circle, and let F( Q) F( xyz,, ) be a contiuously differentiable vector function in a domain containing Then by tokes s theorem and the mean value theorem for surface integrals (see ec 06), Fr( s) ds (curl F) nda (curl F) n( P ) Ar 0, r0 r 0 where the form, A is the area of and P is a suitable point of This may be written in (curl F) n( P ) ( s) ds A Fr r0 In the case of a fluid motion with velocity vector F v the integral vr() s ds, is called the circulation of the flow around It measures the extent to which the corresponding fluid motion is a rotation around the circle Now let approach zero, then, June 0, /6

6 (curl v) n( P) lim v r ( s) ds A, (098) r0 0 r 0 that is, the component of the curl in the positive normal direction can be regarded as the specific circulation (circulation per unit area) of the flow in the surface at the corresponding point Example 5: Work done in the displacement around a closed curve Find the work done by the force F xysin zi xysin zj xycoszk in the displacement around the curve of intersection of the paraboloid cylinder ( x ) y z x y and the olution: This work is given by the line integral in tokes s theorem Now F grad f, where f xy sin z But curl(grad f ) 0 (see eqn (99)) Hence, (curl F) n 0 and the work is 0 by tokes s theorem This agrees with the fact that the present field is conservative (definition in ec 97) 09 tokes's theorem applied to path independence In ec 0 it was noted that, the value of a line integral generally depends not only on the function to be integrated and on the two endpoints A and B of the path of integration, but also on the particular choice of a path from A to B In Theorem of ec 0 it was proved that if a line integral, Fr ( ) dr ( Fdx Fdy Fdz), (099) is path independent in a domain D, then curlf 0 in D It was also claimed in ec 0, that conversely, curl F 0 everywhere in D implies path independence of eqn (099) in D, provided D is simply connected A proof of this needs tokes s theorem, and can now be given as follows Let be any closed path in D ince D is simply connected, we can find a surface in D bounded by tokes s theorem applies and gives F dx F dy F dz F r s ds F nda, ( ) ( ) (curl ) for proper direction on and normal vector n on ince curlf 0 in D, the surface integral and hence the line integral are zero This and Theorem of ec 0 imply that the integral in eqn (099) is path independent in D This completes the proof June 0, /6