(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let F = <-y/(x^{2}+y^{2}, x/(x^{2}+y^{2}>. Recall that F was not conservative on R^{2}- (0,0). In this problem, we show that F is conservative on R^{2}minus the non-positive x-axis. Let D be all of R^{2}except points of the form (-x,0), where x≥0.

a) If (x,y) is included on D, show that it is possible to express (x,y) in polar coordinates (r,θ) such that x=r*cosθ and y=r*sinθ, where θ can be chosen such that θ equals,

{arctan(y)/x if x>0

{pi/2 if x=0

{pi + arctan(y)/x if x<0.

3. The attempt at a solutionIt seems to me that if arctan(y) is bounded between -pi/2 and pi/2, then by taking the limits as x goes to "positive" zero and infinity, we see that θ is bounded, by the first definition, between 0 and infinity.

θ = pi/2 if x=0 simply includes all points along the y-axis.

Similarly, by taking limits, we find that, by the third definition, is bounded between pi and negative infinity. Since (x,y) necessarily lies on D, we need not worry that (x,y) is on the non-positive x-axis.

So what we have here is a θ capable of ranging from negative infinity to infinity. If we choose an arbitrary r, then our range of θ will allow us to reach any (x,y).

Personally, I think that this is either wrong or that there is more convincing way of doing the problem.

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# Polar Coordinates and Conservative Vector Fields

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