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 NewtonLeibniz’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...
View
Full
Document
This document was uploaded on 02/10/2014.
 Spring '13

Click to edit the document details