Lecture 20 Line integral represents the work done by the force V along C W F dr M x N y F T s Now F is conservative When C goes from A to be we get the fundamental theorem of calculus for line integrals Proving this we can use the chain rule Implications Path independent Closed curve Starts and ends at the same point IF C is closed Example f x y C t t C t t where 0 t 1 Three equivalent properties 1 F r 0 closed C 2 path independence for F r F r if c and c have the same endpoints 3 F f for some f Meaning that 1 implies 2 2 implies 3 We must nd f and we want c x t 0 t y F M N f together we get that We know 3 implies 1 from the fundamental theorem of calculus for line integrals How do you know if F is conservative F M N f f Recall the equality of mixed partials f f M N this must hold if F is conservative Example F 2y sin2x 3x cos3y M 2 N 3 M N therefore F is not conservative If M N then F is conservative
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