# Extended Discrete Green's Theorem

Extended Discrete Green's Theorem

This discrete Green's theorem (A Discrete Green's Theorem) connects a given function's double integral over a given domain and the linear combination of the values of the function's cumulative distribution function at the corners of the domain. This suggests a natural extension; by partitioning the domain into rectangles and a curvilinear part, we divide the calculation of the function's double integral over the domain into two parts: the integral over the rectangular domain is calculated using the discrete Green's theorem, and the curvilinear part is calculated via the usual double integral. The "sewing" between these two parts is performed using the parameter of tendency, as suggested in the slanted integration method (Slanted Line Integral). The formula stated by this theorem is simply: f(x,y)dxdyF, where F is the slanted line integral of over the edge of the domain .

∫∫

D

(S)

∫

∂D

(S)

∫

∂D

F

D

In this Demonstration you can control the location of the points , the curvature of each edge connecting two adjacent vertices, the curve's orientation, and which of the subcurves is calculated. A green, red, or black vertex or edge indicates a positive, negative, or zero tendency. A blue vertex or edge means that the tendency is not calculated.

L',M',N',O'