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We have the vector field $F(x,y) = (-\frac y{x²+y²}, \frac x{x²+y²})$

We have three sets:

$V = \{(x,y) ∈ ℝ^2 : y > 0\}$

$W = \{(x,y) ∈ ℝ^2 : y < 0\}$

$T = \{(x,y) ∈ ℝ^2 : x > 0, y = 0\}$

We have the set $U = V ∪ W ∪ T$

I need to find the potentials for $ F$ restricted to $V$, $ F$ restricted to $W,$ and $F$ restricted to $U.$ I found the first two:

$F|_V = \arctan(y/x)$

$F|_W = -\arctan(y/x)$

But could anyone help for the last one?

HeroZhang001
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Arvin
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    Your first two solutions are not defined for $x=0$ which is in $V$ and $W,.$ I think you are better off writing $F={\rm sign}(y)\arccos(x/\sqrt{x^2+y^2})$ which covers all three cases. This answer is related. – Kurt G. Mar 15 '23 at 05:17
  • No I had asked a TA of my course, they had said it's alright the way it is – Arvin Mar 15 '23 at 12:19
  • If $\arctan(y/x)$ is defined on $V$ how about the case $x=0,y>0,?$ – Kurt G. Mar 15 '23 at 13:00

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