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Unformatted text preview: Lecture XXVI The Divergence Theorem In this lecture, we will define a new type of derivative for vector fields on E 3 , called divergence . Let F be a vector field defined on a domain D . Let us start by defining the divergence of F on interior points of D , i.e. points P such that there exists a sphere of center P and radius a > 0 with its interior contained in D . Definition 1 Let F be a continuous vector field on D in E 3 . Let P be an interior point of D , and let S ( P, a ) be the sphere of center P and radius a , for all a > . The volume of S ( P, a ) is 4 a 3 and the ux of F through S ( P, a ) is 3 F d . Consider the limit S ( P,a ) 3 lim F d. a 0 4 a 3 S ( P,a ) If this limit exists, then it is called the divergence of F at P , and it is denoted by div | P . F Below are two important properties of divergence. F is C 1 on D , then the divergence of 1. Existence: If F exists at every interior point of D ....
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