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Math 251 Section 14.5 Curl and Divergence Definition: For F ( x, y , z ) P ( x, y , z ) i Q ( x, y , z ) j R ( x, y , z ) k , i curl F F x P j y Q k R Q R P Q P i k j z y z x y x z R The magnitude of curl F ( x0 , y0 , z 0 ) is a measure of the speed of rotation about and axis through ( x0 , y0 , z 0 ) with the axis pointing in the direction of curl F ( x0 , y0 , z 0 ) . If curl F ( x0 , y0 , z 0 ) 0 , then F is called irrotational at ( x0 , y 0 , z 0 ) . From 14.3 and 14.4, we have that for a two dimensional vector field with F ( x, y ) P ( x, y ) i Q ( x, y ) j on a simply connected domain D with P and Q having continuous partial derivatives in D then, F is conservative and independent of path if and only if Q P curl F k 0 on D. x y The same is true in three dimensions but requires Stokes’ Theorem of section 14.8. We assume it without proof now. Theorem: F is identically 0 in the simply connected domain, D, if and only if F is conservative in D, always assuming all partial derivatives are continuous in D. Note: One direction, f 0 follows from Clairaut’s Theorem. The other direction follows from Stokes’ theorem. Examples: 1. Is F ( x , y , z ) xz i xy j yz k 2. Is x2 conservative? 2 y2 F ( x , y , z ) ( xy e ) i ( 2 zy e 2 x y) j e 2 y2 k conservative? Divergence of a Vector Field Definition: The divergence of the vector field F ( x, y , z ) P ( x, y , z ) i Q ( x, y , z ) j R ( x, y , z ) k div (F ) F If is the scalar function P Q R . x y z div ( F ) 0, F is called incompressible. If P, Q, and R have continuous 2nd partial derivatives on from Clairaut’s Theorem. R 3 then div ( curl ( F )) 0 . This follows