Chapter 9: Vector Calculus
Section 9.9: Stokes' Theorem
The essence of Stokes' theorem is the integration formula
∫∫S∇×F·N dσ = ∳CF·dr = ∳CF·T ds
with sufficient hypotheses for making the formula valid. For example, the vector field F should have continuously differentiable components. The surface S should be oriented, its unit normal field N consistent with the orientation. In addition, S should be described by piecewise-smooth functions and bounded by a piecewise-smooth, simple closed curve C, itself oriented consistently with the orientation of S.
In general, the surface S is not closed. As such, it is called a capping surface for the bounding curve C. For example, a hemisphere is the capping surface for the circle C, at the "equator" of the hemisphere.
The integrand of the double integral on the left is the flux, through the surface S, of the curl of F. On the right, the tangential component of F is integrated around the bounding curve C, yielding the circulation of F around C. Stokes' theorem balances the fields net vorticity (circulation) flowing through the surface S, and the average circulation of F around the bounding curve C. Vorticity (or twist, rotation) of F on the surface S is measured by the curl of F. Integrating the normal component of the curl of F on the surface S allows local swirls that oppose each other to cancel out, leaving just the uncanceled parts on the boundary curve C to reckon with. This residual vorticity is along the bounding curve C and accumulates along C in the line integral on the right side of the Stokes' formula.
Figure 9.9.1 Cancellation of local rotation
Figure 9.9.1 illustrates how local rotation of the normal components of ∇×F "cancel" where two such vectors are contiguous, leaving just the points along the boundary to accumulate circulation. Thus, the net circulation as measured by the flux of the curl field can be measured by summing the tangential component of F along the bounding curve C.
The flux of the curl of F through a closed surface is necessarily zero. This can be seen by applying the Divergence theorem, written as
∫∫SA·N dσ = ∫∫∫R∇·A dv
∯S∇×F·N dσ, where the double integral is now taken over the closed surface S. Take ∇×F in this flux integral as A in the Divergence theorem, and let the interior of the closed surface S be the region R. The result is
∯S∇×F·N dσ = ∫∫∫R∇·∇×F dv = 0
The rightmost (triple) integral vanishes because the integrand is the divergence of a curl. But the divergence of a curl is necessarily zero because curls don't spread, the distillation of Identity 1 in Table 9.4.1.
Stokes' theorem requires that the normal field N chosen for the surface S be consistent with the orientation of the curve C. This coordination is achieved as follows. Choose a normal direction on S. Then traverse the curve C with the left hand pointing in towards S and the head pointing in the direction of N.
For example, if the curve C is the unit circle centered at the origin, and the capping surface S is the upper hemisphere, then an outward normal on S induces a counterclockwise orientation of the curve C.
Apply Stokes' theorem to the vector field F=z i−x j−y k; the curve C, the unit circle with center at the origin; and the upper hemisphere as the capping surface S.
Show that the flux of the curl of F=z i−x j−y k through the unit sphere centered at the origin is necessarily zero.
Apply Stokes' theorem to the vector field F=y z i+x2z j+x y k; the curve C, the unit circle with center at the origin; and a closed cylinder with lid in the plane z=a>0 as the capping surface S.
Apply Stokes' theorem to the vector field F=x y i+y z j+x z k; the curve C, the triangle with vertices 0,0,20,30,0,0,0,24,0; and the first-octant portion of the plane 4 x+5 y+6 z=120 as the capping surface S.
Apply Stokes' theorem to the vector field F=x y i+y z j+x z k; the curve C, the triangle with vertices 0,0,0,12,0,0,0,8,0; and the first-octant portion of the plane 2 x+3 y+4 z=24 along with the coordinate planes x=y=0 as the capping surface S. Hint: S has three sides but is open at the bottom.
Apply Stokes' theorem to the vector field F=x2 y i+y z j+x z k; the curve C, the ellipse x2+4 y2=1; and the upper half of the ellipsoid x2+4 y2+6 z2=1 as the capping surface S.
Show that the flux of the curl of F=x2 y i+y z j+x z k through S, the upper portion of the ellipsoid x2+4 y2+6 z2=1 closed with the interior and boundary of an ellipse in the plane z=0, is necessarily zero.
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