Properties of a phase plane

It may be shown that Lt is also a smooth line without self-intersections. 

The area of the figure Qt contained by the line Lt expressed by the equation 

We may now compute the change in the area Qt with time by differentiating equation (A5) with respect to time t 

Integrating by parts the second integral in (A6b) and employing the equality ( fi(9/2)/)r =o = 0 we obtain 

Firstly, if in a certain connected region divF is everywhere different from zero (divF 0 or divF 0) then the system (Al) cannot have limit cycles located entirely in this region. This is the so-called Bendixon criterion. 

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