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# lec7-1 - F Β΄ Β Λ k = β F y β x-β F x β y β’ So...

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Green’s theorem in the plane We have seen how to do double integrals, now let’s do them in the xy-plane over an area A Z b a dx Z y u y l P ( x , y ) y dy = Z b a [ P ( x , y u ) - P ( x , y l )] dx = - I C Pdx The integral H C Pdx means a counterclockwise integral around the curve that bounds area A We could also do the same thing but integrate first with respect to y Chapter7: Fourier series

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Continued, Green’s theorem in the plane Z d c dy Z x r x l Q ( x , y ) x dx = Z d c [ Q ( x r , y ) - Q ( x l , y )] dy = - I C Qdy Putting these together we get Green’s theorem in the plane, Z Z A Q x - P y dxdy = I A ( Pdx + Qdy ) Chapter7: Fourier series
Green’s theorem in the plane, connection to line integrals and work Z Z A Q x - P y dxdy = I A ( Pdx + Qdy ) If we take F x = P and F y = Q , then we see Green’s theorem in the plane gives Z Z A F y x - F x y dxdy = I A ( F x dx + F y dy ) = W Then notice that ∇ ×

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Unformatted text preview: F Β΄ Β· Λ k = β F y β x-β 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 x-β P β y = β V x β x + β V y β y = div ~ V β’ Consider the outward normal Λ n = Λ idx-Λ jdy β dx 2 + dy 2 = Λ idx-Λ jdy ds , and then Pdx + Qdy =-V y dx + V x dy = ( V x Λ i + V y Λ j ) Β· ( Λ idy-Λ jdx ) = ~ V Β· Λ nds β’ The by Greenβs theorem in the plane, Z Z A div ~ V dxdy = Z β A ~ V Β· Λ nds Chapter7: Fourier series...
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