I want to check a possible solution to a problem I cannot solve:
We know that closed 2-forms for $R^2-\{p,q\}$ are exact. Given a closed 2-form, what is a 1-form that gives us that 2-form under exterior derivative?
For $R^2-\{p\}$, just one point removed, we can do so in polar coordinates:
ω=f(r,θ)dr∧dθ. Then $h=(\int_1^r f(u,θ)du)dθ$ is a primitive of ω.
for $R^2-\{p,q\}$, could I consider two tangent circles of equal radius centered at each point and integrate points inside each circle just like the above, and on the outside sum the integrals?
Can anyone give the computation?
(this is not an assigned HW problem, just self reading a book)
Anyone ?