# Trying to assemble an indispensable enhancement theorem

Let $C$ be an aircraft contour offered by the set $\{(x,y) \in \mathbb{R}^2: \, P(x,y) = 0 \}$ and also specify $\omega=\frac{\mathrm{d}x}{y}$. After that, it is

$$\int\limits_0^A \omega + \int\limits_0^B \omega = \int\limits_0^{A \oplus B} \omega$$

(with $\oplus$ being enhancement on a team on the contour ) a theory?

I'm fairly certain this is what's taking place when C is an elliptic contour, yet I have actually fallen short to make this exercise when C is specified by $P(x,y) = x^2 + y^2 - 1$ (device circle ) and also the team regulation is specified by shooting a ray via $(3/5,4/5)$ (picked randomly ) alongside $AB$ and also taking it's junction with the circle as $A \oplus B$.

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2019-05-07 10:49:26
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$0$, the lower restriction for the integrals, need to be whatever factor on the contour is $0$ for the team regulation.

After that $dx/y$ needs to be turning - stable for any kind of circle. It will certainly not hold true for a revolved ellipse ; there is a stable differential yet it is not $dx/y$.

For an elliptic contour, $dx/y$ being the stable differential relies upon the contour being created as $y^2=f(x)$ with $\deg(f)=3$. If you relocate the contour the stable differential will certainly be a various one.

For greater categories the contour does not have an enhancement regulation.

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2019-05-09 05:11:53
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