14.3 Notes - t x t r = , ) ( ), ( ) ( . If j N i M y x F +...

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Calculus III Section 14.3 – Conservative Vector Fields and Independence of Path Example: Find the work done by the force field ( 29 ( 29 j xy i y y x F 2 1 ) , ( 2 + + = on a particle that moves from (0,0) to (2,0) along each path: a) b) c)
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Theorem – Independence of Path and Conservative Vector Fields If F is continuous on an open connected region, then the line integral C r d F is independent of path if and only if F is conservative. Example: #16
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Fundamental Theorem of Line Integrals Let C be a piecewise smooth curve lying in an open region R and given by b t a t y
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Unformatted text preview: t x t r = , ) ( ), ( ) ( . If j N i M y x F + = ) , ( is conservative in R , and M and N are continuous in R , then ( 29 ( 29 ) ( ), ( ) ( ), ( a y a x f b y b x f r d f r d F C C-= = where f is a potential function of F. That is, ) , ( ) , ( y x f y x F = . Example: Use the Fundamental Theorem of Line Integrals to evaluate C r d F where ( 29 ( 29 j xy i y y x F 2 1 ) , ( 2 + + = along the path from (0,0) to (2,0) as shown in the diagram. How cool is that?!...
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14.3 Notes - t x t r = , ) ( ), ( ) ( . If j N i M y x F +...

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