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16-3 - Math 200 Spring 2010 Handout#25 Section 16-3...

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Math 200, Spring 2010 Handout #25 Section 16-3 Conservative Vector Fields Independence of Path and Conservative Vector Fields We say that the integral C d F s is independent of path if the integral is the same for every path having the same initial and terminal points, that is, 1 2 d d = c c F s F s for any two paths ( ) 1 t c and ( ) 2 t c from point P to point Q . A vector field F with this property is called conservative . Definition: A curve is closed if its two endpoints are the same. Theorem: For any closed path C in domain D , C d F s is path independent in D if and only if 0 C d = F s integralloop . Note: This theorem says that the work done by a conservative vector field moving an object along a closed curve is 0. The Fundamental Theorem for Gradient Vector Fields If ϕ = ∇ F on a domain D , then for every oriented curve C in D with initial point P and terminal point Q , ( ) ( ) C C d d Q P ϕ ϕ ϕ = = F s s . Note the similarity to the Fundamental Theorem of Calculus, which gives the integral of a derivative as the original function evaluated at the endpoints of an interval: ( ) ( ) ( ) b a f x dx f b f a = . Since the gradient is just another type of derivative, the above theorem should seem plausible. Note: This theorem says that if a vector field F is a gradient vector field, then we have a nice and easy approach for computing line integrals, and thus the work done by a conservative force field. Note: This also says that the line integral of a gradient vector field depends only on the endpoints of the curve C , not the path taken, that is, it is path independent. Note: If C is closed and F is a gradient vector fields, then 0 C d = F s integralloop 1. Let F be the gradient of ( ) 2 , , x y z xy z ϕ = + . Compute the line integral of F along: (a) The line segment from ( ) ( ) 1,1,1 to 1,2,2 .
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