Stokes Theorem

# Stokes Theorem - SECTION 16.5 Stokes's Theorem 875 16.5...

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SECTION 16.5: Stokes’s Theorem 875 16.5 Stokes’s Theorem If we regard a region R in the xy -plane as a surface in 3-space with normal Feld ˆ N = k , the Green’s Theorem formula can be written in the form I C F d r = Z R curlF ˆ N dS . Stokes’s Theorem given below generalizes this to nonplanar surfaces. THEOREM 10 Stokes’s Theorem Let S be a piecewise smooth, oriented surface in 3-space, having unit normal Feld ˆ N and boundary C consisting of one or more piecewise smooth, closed curves with orientation inherited from S .I f F is a smooth vector Feld deFned on an open set containing S ,then I C F d r = Z S curl F ˆ N . PROOF An argument similar to those given in the proofs of Green’s Theorem and the Divergence Theorem shows that if S is decomposed into Fnitely many nonoverlapping subsurfaces, then it is sufFcient to prove that the formula above holds for each of them. (If subsurfaces S 1 and S 2 meet along the curve C C inherits opposite orientations as part of the boundaries of S 1 and S 2 , so the line integrals along C cancel out. See ±igure 16.15(a).) We can subdivide S into enough smooth subsurfaces that each one has a one-to-one normal projection onto a coordinate plane. We will establish the formula for one such subsurface, which we will now call S . Figure 16.15 (a) Stokes’s Theorem holds for a composite surface comprised of nonoverlapping subsurfaces for which it is true (b) A surface with a one-to-one projection on the -plane C S 2 S 1 C ˆ N x y z k k ˆ N z = g ( x , y ) C C R S (a) (b) Without loss of generality, assume that S has a one-to-one normal projection onto the -plane and that its normal Feld ˆ N points upward. Therefore, on S , z is a smooth function of x and y ,say z = g ( x , y ) , deFned for ( x , y ) in a region R of the -plane. The boundaries C of S and C of R are both oriented counterclockwise as seen from a point high on the z -axis. (See ±igure 16.15(b).) The normal Feld on S is ˆ N = g x i g y j + k s 1 + g x ± 2 + g y ± 2 , and the surface area element on S is expressed in terms of the area element dA = dx dy in the -plane as = s 1 + g x ± 2 + g y ± 2 .

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876 CHAPTER 16 Vector Calculus Therefore, ±± S curl F ˆ N dS = R ²³ F 3 y F 2 z ´³ g x ´ + ³ F 1 z F 3 x g y ´ + ³ F 2 x F 1 y ´µ dA . Since z = g ( x , y ) on C ,wehave dz = g x dx + g y dy . Thus, C F d r = C ² F 1 ( x , y , z ) + F 2 ( x , y , z ) + F 3 ( x , y , z ) ³ g x + g y ´µ = C ³² F 1 ( x , y , z ) + F 3 ( x , y , z ) g x µ + ² F 2 ( x , y , z ) + F 3 ( x , y , z ) g y µ ´ . We now apply Green’s Theorem in the xy -plane to obtain C F d r = R ³ x ² F 2 ( x , y , z ) + F 3 ( x , y , z ) g y µ y ² F 1 ( x , y , z ) + F 3 ( x , y , z ) g x µ´ = R ³ F 2 x + F 2 z g x + F 3 x g y + F 3 z g x g y + F 3 2 g x y F 1 y F 1 z g y F 3 y g x
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## This note was uploaded on 03/21/2012 for the course STAT 101 taught by Professor Graham during the Spring '08 term at Iowa State.

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Stokes Theorem - SECTION 16.5 Stokes's Theorem 875 16.5...

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