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May 30, 2015 at 13:36 comment added Michael E2 @chris I think in general the answer is (-1)^n, where n is the dimension. The -1 because the path is clockwise instead of counterclockwise (which I think you referred to in the Q).
May 30, 2015 at 13:05 comment added chris Do you understand the -1 in the triple integral case?
May 30, 2015 at 11:03 history edited Michael E2 CC BY-SA 3.0
Corrected explanation
May 30, 2015 at 10:42 vote accept chris
May 30, 2015 at 4:06 history edited chris CC BY-SA 3.0
added 438 characters in body
May 30, 2015 at 3:54 comment added chris Very nicely spotted! I had wrongly convinced myself this was a fundamental problem, not a bug.
May 30, 2015 at 3:36 history answered Michael E2 CC BY-SA 3.0