Скачать или смотреть #08 Green's theorem in xy plane شرح نظرية جرين

Green's theorem is a fundamental result in vector calculus that relates a line integral around a simple closed curve in the xy-plane to a double integral over the region enclosed by that curve. It establishes a connection between the circulation around a curve and the flux across the enclosed region.

In the two-dimensional Cartesian coordinate system (xy-plane), Green's theorem states:

∮C F · dr = ∬D ( ∂Q/∂x - ∂P/∂y ) dA

where:

∮C represents the line integral taken counterclockwise around the closed curve C.
F = (P, Q) is a vector field defined on the region D enclosed by C, where P and Q are functions of x and y.
dr is an infinitesimal vector tangent to the curve C.
∂Q/∂x and ∂P/∂y are the partial derivatives of Q with respect to x and P with respect to y, respectively.
∬D represents the double integral over the region D enclosed by C.
dA is an infinitesimal area element in the xy-plane.
In simpler terms, Green's theorem states that the line integral of a vector field F around a closed curve C is equal to the double integral of the curl of F over the region D enclosed by C. The curl of F is given by ∂Q/∂x - ∂P/∂y, where P and Q are the components of the vector field.

Green's theorem is a powerful tool in physics and engineering, particularly in the study of fluid flow, electromagnetism, and potential theory. It provides a useful relationship between line integrals and surface integrals, allowing for the conversion of one type of integral into the other when solving various problems involving vector fields in the xy-plane.