Applications of Stoke's Theorem

Suppose two surfaces S₁ and S₂ share a common boundary curve C. If a vector field F( x,y,z) is sufficiently differentiable, then Stoke's theorem implies that

ó õ ó õ S1 curl( F) ·dS =   C  F·dr =  ó õ ó õ S2 curl( F)·dS

This is useful because the flux of the curl of F is often more easily calculated on one surface than on the other. Equivalently, it says that we can compute the circulation of F around a closed curve C using any surface S that has C as its boundary.      

EXAMPLE 4    Choose an appropriate surface and apply Stoke's theorem to

T  -ydx+xdy

Solution: If F( x,y,z) = á-y, x, 0 ñ , then

T  ydx-xdy = T  F · dr

Notice that T is the boundary of the upper unit hemisphere.

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Thus, if S denotes the upper unit hemisphere, then

T  F·dr =   ó õ ó õ S curl( F) · dS

However, equation (3) in section 5-5 implies that

ó õ ó õ S curl( F) ·dS =  ó õ 2p 0  ó õ p/2 0  curl( F) · áx,y,z ñ   sin( f) dfdq

where x = sin( f) cos( q) , y = sin( f) sin( q) , and z = cos( f) . Straightforward calculation yields curl( F) = á 0,0,2 ñ , so that

ó õ ó õ S curl( F) ·dS = ó õ ó õ S   á 0,0,2 ñ · áx,y,z ñ sin( f) dfdq = ó õ 2p 0  ó õ p/2 0  2cos( f) sin(f)   dfdq = ó õ 2p 0  ó õ p/2 0  sin( 2f)   dfdq = ó õ 2p 0   -1 2 cos( 2f) ê ê p/2 0  dq = ó õ 2p 0  dq = 2p

This matches the result in example 1, where we considered the same line integral. Notice also that we also obtained the same flux through the paraboloid in example 1 as we did through the unit upper hemisphere here. That is because both surfaces share the same boundary-i.e., the unit circle T.

As another application, let's use Stoke's theorem to find the area of a region S which is in a plane with a normal N = á a,b,c ñ. The unit normal of the plane is n = N/|| N|| , and if we let F( x,y,z) = ábz,cx,ay ñ , then curl  F = áa,b,c ñ and

( curl F) ·n = N·  N || N|| = || N|| =   a2+b2+c2

Since curl( F) · dS = curl( F) · n  dS, Stoke's theorem reduces to

ó õ ó õ S a2+b2+c2   dS = ¶S  á bz,cx,ay ñ · ádx,dy,dz ñ

Since the area of the region S is ∫∫_S dS, the area of a region S in a plane with normal n = á a,b,c ñ is given by

Area  of  S = ê ê ê  1 a2+b2+c2 ¶S  bzdx+cxdy+aydz ê ê ê

(1)

This allows us to compute the area inside of a closed curve in an arbitrary plane.       

EXAMPLE 5    Show that r( t) = á4sin( t), 5cos( t), 3sin( t) ñ , t in [ 0,2p] , is a closed curve in a plane by showing that n = r×v has constant direction. Then find the area enclosed by that curve in the plane with normal n.