{"id":5733,"date":"2014-09-15T06:05:05","date_gmt":"2014-09-14T21:05:05","guid":{"rendered":"http:\/\/fujiitoshiki.com\/improvesociety\/?p=5733"},"modified":"2014-08-08T19:44:46","modified_gmt":"2014-08-08T10:44:46","slug":"greens-theorem-in-the-plane","status":"publish","type":"post","link":"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=5733","title":{"rendered":"GREEN’S THEOREM IN THE PLANE"},"content":{"rendered":"
\n

Let P,\\ Q,\\ \\partial P\/\\partial y,\\ \\partial Q\/\\partial x be single-valued and continuous in a simply-connected region \\cal R bounded by a simple closed curve C. Then <\/p>\n

\\displaystyle \\oint_C [Pdx + Qdy] = \\underset{\\cal R}{\\iint}\\left( \\frac{\\partial Q}{\\partial x} - \\frac{\\partial P}{\\partial y} \\right)dxdy\\cdots(22)<\/p>\n

where \\oint_C is used to emphasize that C is closed and that it is described in the positive direction. <\/p>\n

This theorem is also true for regions bounded by two or more closed curves (i.e. multiply-connected regions). <\/p>\n<\/blockquote>\n