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Unformatted text preview: 16.8: STOKES’ THEOREM KIAM HEONG KWA 1. Stokes’ Theorem Let S be an oriented piecewise smooth surface whose boundary C is a simple, closed, piecewise smooth curve. Let ( x,y,z ) 7→ n ( x,y,z ) be the given orientation of S and let r ( t ) = x ( t ) i + y ( t ) j + z ( t ) k , a ≤ t ≤ b, be a parametrization of C . Then C is said to have the induced positive orientation as the boundary curve of S if at each point (¯ x, ¯ y, ¯ z ) = ( x ( ¯ t ) ,y ( ¯ t ) ,z ( ¯ t )) of C at which r ( ¯ t ) exists and is nonzero, the cross product n (¯ x, ¯ y, ¯ z ) × r ( ¯ t ) always points inward in the direction of the surface S . It is customary to denote C by ∂S whenever it has the induced positive orientation. Theorem 1 (Stokes’ Theorem) . If F is a vector field whose compo- nents have continuous partial derivatives on an open region in R 3 that contains an oriented piecewise smooth surface S , then (1.1) Z ∂S F · d r = ZZ S ∇ × F · d S . There are a few interesting corollaries of Stokes’ theorem: (1) If S 1 and S 2 are two oriented piecewise smooth surfaces such that ∂S 1 = ∂S 2 , then ZZ S 1 ∇ × F · d S = ZZ S 2 ∇ × F · d S since the surface integrals are both equal to the line integral of F along ∂S 1 = ∂S 2 by Stokes’ theorem....
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