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Unformatted text preview: F Β΄ Β· Λ k = β F y β xβ F x β y β’ So if β ~ F = 0, then W = 0 as we already know Chapter7: Fourier series Divergence theorem in two dimensions β’ Consider a vector ο¬eld ~ V = V x Λ i + V y Λ j (Notice here V z = 0) β’ If we take Q = V x and P =V y , then β Q β xβ P β y = β V x β x + β V y β y = div ~ V β’ Consider the outward normal Λ n = Λ idxΛ jdy β dx 2 + dy 2 = Λ idxΛ jdy ds , and then Pdx + Qdy =V y dx + V x dy = ( V x Λ i + V y Λ j ) Β· ( Λ idyΛ jdx ) = ~ V Β· Λ nds β’ The by Greenβs theorem in the plane, Z Z A div ~ V dxdy = Z β A ~ V Β· Λ nds Chapter7: Fourier series...
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 Spring '03
 Staff
 Fourier Series, Flux, Stokes' theorem, Qdy

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