Total amount of particles crossing s F n as s approaches 0 Flux F n s Lecture 23 Vector Field F M N velocity of uid How much uid ows out Which particles cross The ones where the arrows cross the boundary C The vector eld is almost constant in a small region Area s F n Example Example C unit circle around 0 F x y F n 1 Flux 2 F y x F n 0 Flux 0 Computing using Green s Theorem Flux F n s We need to write this using F r or F T s or P x Q y For n s C x t y t and r x y t x y The unit normal n n s n s y x t y x Flux F n s M N y x N x M y curl N M A M N A De nition When the vector eld is M N the divergence is div F M N The Green Theorem version of ux F n s div F A If the divergence is zero the ux is zero Example F 2x 3y Area of a theta wedge Flux 5 A 5 6 30 Div x y 2 2Area x y n s Length of circular chord of angle 0 0 The Laplacian where u is a function on R u div u div u u u u If u is zero u is harmonic
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