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cal18 - Chapter Eighteen Stokes 18.1 Stokes's Theorem Let F...

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18.1 Chapter Eighteen Stokes 18.1 Stokes's Theorem Let F R 3 : D be a nice vector function. If F i j k ( , , ) ( , , ) ( , , ) ( , , ) x y z p x y z q x y z r x y z = + + , the curl of F is defined by curl r y q z p z r x q x p y F i j k = - + - + - . Here also the so-called del operator ∇ = + + i j k x y z provides a nice memory device: curl x y z p q r F F i j k = ∇ × = . This definition allows us to look at Green's Theorem from a new perspective by observing that in case F i j ( , ) ( , ) ( , ) x y p x y q x y = + , Green's Theorem becomes ( ) F r F S = ∫∫ d curl d R C , where we are thinking of the region R as an oriented surface with its orientation pointing in the direction of k . We want to look at this formula in case the region R is not necessarily in the i-j plane, in which case, the word "clockwise" doesn't help in deciding on the orientation of the boundary C . Once again, we orient things according to our familiar "right-hand" rule.
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18.2 Here's the way it goes. Suppose now S is any surface bounded by a finite number of disjoint curves C C C n 1 2 , , K . We say simply that C C C C n = 1 2 K is the boundary of S. Now choose an orientation for the surface S . Look at one of these normal vectors "close" to a curve C j and imagine a little circle around the base of the normal oriented so that the normal vector points in the right-hand direction with respect to the direction of the circle. Then the orientation, or direction, of C j that is consistent with the given orientation of the surface S is the one that "lines up" with the direction on this little circle.
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