(adsbygoogle = window.adsbygoogle || []).push({}); Analyzing an integral over a non-exact region for gamma defined by |x|+|y|=4

The following was similar to a problem on a calculus final that I got wrong. It is an extension of a problem in R.C. Buck "Advanced Calculus" on page 501. Similar to knowing the trick to integrating [itex] e^{|x|}[/itex] (which got me on my first calculus final years ago!), I assume there is a bit of analysis that I'm missing for this line integral regarding the absolute value. Someone please walk me through the analysis of this problem from Buck taken a step further.

Consider the 1-form [itex]\omega = \frac{xdy-ydx}{x^{2}+y^{2}}[/itex] in the open ring [itex]D = \lbrace (x,y) | 1 \leq x^{2} + y^{2} \leq 4 \rbrace[/itex].

The text asserts that [itex]d \omega[/itex] is not exact in D, which I verified with the standard computation.

The extension is -- setting [itex] \gamma (t)[/itex] to be [itex]|x| + |y| = 4[/itex], calculate the integral [itex]\int_{\gamma}\omega[/itex] counterclockwise over the region.

My hangups: aside from the differential form not being exact, the curve is closed, but not smooth.

Please note I don't necessary want someone to do this problem. I'd much rather have someone discuss how to pick this problem apart analytically. What is the first question to ask yourself? How do you reconcile the curve not being smooth? etc.

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# Analyzing the line integral over a non-exact region for gamma defined by |x|+|y|=4

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