Hence that vector eld is not conservative 4 j i 2 2

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Unformatted text preview: +2 2 j− i = 2π 2+ 2 In part (b) we see that there are closed curves for which the line integral of non zero. Hence that vector field is not conservative. 4 j− i 2+ 2 is :) 4 Extra examples on Newton-Leibniz’s and Green’s theorems E + Evaluate 2 C + 2 , where C is some crazy path initiating at (1 0) and terminating at (6 8). S Let’s see if we can find a potential function potential must satisfy =2 = 2+ 2 + for the vector field. Such a 2 Either by calculating or by inspection, we see that ( quirement, so + 2 C E + 2 = (6 8) − (1 1) = 2 )= + 2 satisfies that re- √ √ 36 + 64 − 1 + 0 = 10 − 1 = 9 Evaluate the integral 2 − 2 −− 2 C 2 where C is the circle S 2 + = 2 . By Green’s theorem, we have 2 C − 2 2 = = ( 2+ 2≤ 2 2 + 2 ) 2+ 2≤ 2 In polar coordinates, the above integral can be evaluated as 2π ( 2...
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