Green's theorem and area calculation

[Jhenrique]

In wiki there is the follows formula:

https://en.wikipedia.org/wiki/Green's_theorem#Area_Calculation

But, I don't understand why M = x and L = -y. I don't found this step in anywhere.

 

[SteamKing]

In wiki there is the follows formula:

https://en.wikipedia.org/wiki/Green's_theorem#Area_Calculation

But, I don't understand why M = x and L = -y. I don't found this step in anywhere.

Go back to the very definition of the Green's Theorem in the 'Theorem' section of the article. You pick M and N arbitrarily so that the equivalent double integral for the area of the region enclosed by the curve C can be expressed as a line integral around C. These choices of M and N are not necessarily unique.

 

[LCKurtz]

In wiki there is the follows formula:

https://en.wikipedia.org/wiki/Green's_theorem#Area_Calculation

But, I don't understand why M = x and L = -y. I don't found this step in anywhere.

If $\vec F = \langle P(x,y),Q(x,y)\rangle$ then Green's theorem is$$
\iint_R Q_x - P_y~dA =\oint_C \vec F\cdot d\vec r$$If $Q_x - P_y = 1$ the double integral on the left gives the area of the enclosed region $R$. Any choices of $P$ and $Q$ that give $1$ will work. The choices in your example are simple examples that work but you could likely find others.