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Unformatted text preview: Lecture XXV Green’s Theorem Let us define a new type of derivative, called rotational derivative , applicable to vector fields in E 2 . For such a vector field F on a domain D in E 2 , let us define the rotational derivative at interior points of D . Here, a point P in D is called an interior point if there exists a circle of center P and radius a > 0 that has its interior contained in D . Definition 1 Let F be a C 1 vector field on D in E 2 . Let P be an interior point of D and let C ( P, a ) be the circle of center P and radius a for all a > . The rotational derivative, denoted by rot P , is given by the following formula: F | rotF 1 d F R. = lim · P a → πa 2 C ( P,a ) F | For boundary points Q of D , if lim P → Q,P rot P = q exists, we say that rot Q F | F is C 1 , then rot | P exists at every point P on the exists and its value is q . If F domain D of F . The rotational derivative satisfies the linearity law: G ) = a ( rot G ) . rot ( aF + b F ) + b ( rot Theorem 1 (The parallel ﬂow theorem in E...
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