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# 16-2b - Math 200 Spring 2010 Handout 24 Section 16-2b...

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Math 200, Spring 2010 Handout # 24 Section 16-2b Vector Line Integrals Orientation of a Curve C Since a curve C can be travered in one of two directions, we say C is oriented if one of these two directions is specified, and the direction specified is referred to as the forward direction. If C denotes the curve consisting of the same points as C but with opposite orientation, then we have ( ) ( ) , , C C f x y dx f x y dx = − Vector Line Integral The line integral of a continuous vector field F along an oriented curve C is the tangential component of the vector field along the curve and is denoted by C d F s i . Conceptually, the vector line integral is a limit of Riemann sums: Divide curve C into subarcs 1 i i P P with lengths i s Δ , choose a point ( ) * * * * , , i i i i P x y z on the i th subarc. Let ( ) * * * , , i i i x y z T denote the unit tangent vector pointing in the forward direction along the curve. Then ( ) ( ) * * * * * * , , , , i i i i i i x y z x y z F T i represents the tangential component of F at * i P on C . The sum of these tangential components of F form a Reimann sums ( ) ( ) * * * * * * 1 , , , , i i i i i i i n i x y z x y z s = Δ F T i . for a vector field F of three variables whose domain includes the curve C . The line integral of F along C is the limit (if it exists) of these Riemann sums as the maximum of the lengths i s Δ approaches zero: { } ( ) ( ) ( ) ( ) * * * * * * 0 1 , , , lim , , , , , C C i i i i i i i C i n s i x y z d x y z x y z x y z ds ds s Δ = = = Δ =

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16-2b - Math 200 Spring 2010 Handout 24 Section 16-2b...

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