see

https://youtu.be/l5zJvZKfMYE

https://youtu.be/qdFD-0OWBRo

https://youtu.be/gGXnILbrhsM

https://youtu.be/sSyPAAyL8nQ

The Green theorem is a special case of the Stokes theorem which is a main tool for understanding conservative vector fields.

This is an interesting question. Let us first give a formulation of Green's theorem.

Let C be a positively oriented, piecewise smooth, simple closed curve in a plane, and let D be the region bounded by C. If L and M are functions of (x, y) defined on the closure of an open region containing D and have continuous partial derivatives on the closure, then

\int_ C ( L d x + M d y ) = \int_ D ( M'_x - L'_ y ) d x d y {\displaystyle \oint _{C}(L\,dx+M\,dy)=\iint _{D}\left({\frac {\partial M}{\partial x}}-{\frac {\partial L}{\partial y}}\right)\,dx\,dy}

where the path of integration along C is counterclockwise.

In physics, Green's theorem is mostly used to solve two-dimensional flow integrals, stating that the sum of fluid outflows from a volume is equal to the total outflow summed about an enclosing area. In plane geometry, and in particular, area surveying, Green's theorem can be used to determine the area and centroid of plane figures solely by integrating over the perimeter.

See Mathematical methods for physics and engineering, K.F. Riley, M.P. Hobson, S.J. Bence, Cambridge University Press, 2010, ISBN 978-0-521-86153-3

Vector Analysis (2nd Edition), M.R. Spiegel, S. Lipschutz, D. Spellman, Schaum’s Outlines, McGraw Hill (USA), 2009, ISBN 978-0-07-161545-7

As soon as possible I will give more details.