Assume that is an increasing continuously dierentiable

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Unformatted text preview: t change if we substitute = ( ). Assume that = ( ) is an increasing continuously differentiable function from [α β] to [ (increasing because line integral of a vector field depends on the orientation). Then · = = and P =1 P = P = =1 =1 =1 so it is the same as the original expression in . 4 :) 4 Extra examples on vector fields and line integrals E Evaluate 2 2 + C 2 where C is the circle + 2 = . S The circle can be rewritten as ( − /2)2 + eterize it by = = + 2 2 sin 2 2 = ( /2)2 . Hence we can param- cos 0 ≤ < 2π In this case, we have 2 2 + = = (− cos )2 + sin2 2 = 2 Thus, we have 2π 2 + 2 2 = C 2 0 2π 2 = 2 2 √ 2 cos 0 2 2 2 = 2π 2 4 cos2 0 2 π | cos | 2 = 2 | cos | 0 π − cos cos =2 2 π/2 0 E sin 2 2 + 2 cos π /2 = 2 + 0 2π 2 =2 + Evaluate C + | |+| | where C is the square with vertices (1 0), (0 1), (−1 0), (0 −1) (in this order). 5 :...
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This document was uploaded on 02/10/2014.

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