From the defintion of a closed Line Integral we know that if the F (vector) is conservative that the integral will = 0, and will always be indipendent of path.
What if, however, we have a closed line integral along a curve symmetric about the X and Y axes?
Sorry, i don't know how to make neat equations like other posters.
Evaluate the closed Integral of (y^2dx + x^2dy) clockwize about
the curve |x| + |y| = 1
Clearly this has 4 cases in the XY domain.
Case 1: x + y = 1 => y = 1-x => dy = -dx
Case 2: x - y = 1 => y = x-1 => dy = dx
Case 3: -x - y = 1 => y = -x-1 => dy = -dx
Case 4: -x + y = 1 => y = 1 + x => dy = dx
I wont bother going through all the line integrals with you. But I do the line integrals with each case with respect to X.
So the X bounds for:
Case 1: 0-->1
Case 2: 1-->0
Case 3: 0-->-1
Case 4: -1--> 0
Then I compute and add them up and it equals to 0. This promted this question because my intial function is not conservative. So I must ask is a closed line integral along a curve symmetric about the X and Y axes = 0? Or did I compute this incorrectly somewhere down the road?