ch14_08 - P1 OSO/OVY GTBL001-14-nal P2 OSO/OVY QC OSO/OVY...

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14-85 SECTION 14.8 . . Stokes’ Theorem 1199 14.8 STOKES’ THEOREM Recall that, after introducing the curl in section 14.5, we observed that for a piecewise, smooth, positively oriented, simple closed curve C in the xy -plane enclosing the region R , we could rewrite Green’s Theorem in the vector form ± C F · d r = ²² R ( ∇× F ) · k dA , (8.1) where F ( x , y )isavector field of the form F ( x , y ) M ( x , y ) , N ( x , y ) , 0 ² .In this section, we generalize this result to the case of a vector field defined on a surface in three dimensions. Suppose that S is an oriented surface with unit normal vector n .If S is bounded by the simple closed curve C ,we determine the orientation of C using a right-hand rule like the one used to determine the direction of a cross product of two vectors. Align the thumb of your right hand so that it points in the direction of one of the unit normals to S . Then if you curl your fingers, they will indicate the positive orientation on C ,as indicated in Figure 14.52a. If the orientation of C is opposite that indicated by the curling of the fingers on your right hand, as shown in Figure 14.52b, we say that C has negative orientation. The vector form of Green’s Theorem in (8.1) generalizes as follows. S C z n y x FIGURE 14.52a Positive orientation S C z y x n FIGURE 14.52b Negative orientation THEOREM 8.1 (Stokes’ Theorem) Suppose that S is an oriented, piecewise-smooth surface with unit normal vector n , bounded by the simple closed, piecewise-smooth boundary curve S having positive orientation. Let F ( x , y , z )beavector field whose components have continuous first partial derivatives in some open region containing S . Then, ² S F ( x , y , z ) · d r = S ( F ) · n dS . (8.2) Notice right away that the vector form of Green’s Theorem (8.1) is a special case of (8.2), as follows. If S is simply a region in the xy -plane, then a unit normal to the surface at every point on S is the vector n = k . Further, dS = dA (i.e., the surface area of the plane region is simply the area) and (8.2) simplifies to (8.1). The proof of Stokes’ Theorem for the special case considered below hinges on Green’s Theorem and the chain rule. One important interpretation of Stokes’ Theorem arises in the case where F represents a force field. Note that in this case, the integral on the left side of (8.2) corresponds to the work done by the force field F as the point of application moves along the boundary of S . Likewise, the right side of (8.2) represents the net flux of the curl of F over the surface S . A general proof of Stokes’ Theorem can be found in more advanced texts. We prove it here only for a special case of the surface S . PROOF(Special Case) We consider here the special case where S is a surface of the form S ={ ( x , y , z ) | z = f ( x , y ) , for ( x , y ) R } ,
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1200 CHAPTER 14 . . Vector Calculus 14-86 where R is a region in the xy -plane with piecewise-smooth boundary R , where f ( x , y ) has continuous first partial derivatives and for which R is the projection of the boundary of the surface S onto the xy -plane (see Figure 14.53).
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This note was uploaded on 10/04/2010 for the course CHE2C 929102 taught by Professor Carter during the Spring '10 term at UC Davis.

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ch14_08 - P1 OSO/OVY GTBL001-14-nal P2 OSO/OVY QC OSO/OVY...

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