# lec7-1 - F k = F y x- F x y So if ~ F = 0, then W = 0 as we...

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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 yu 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 xr 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 ² 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 ² 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 ² dxdy = I A ( F x dx + F y dy ) = W Then notice that ³ ∇ × ~

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Unformatted text preview: F k = F y x- F x y So if ~ F = 0, then W = 0 as we already know Chapter7: Fourier series Divergence theorem in two dimensions Consider a vector eld ~ V = V x i + V y j (Notice here V z = 0) If we take Q = V x and P =-V y , then Q x- P y = V x x + V y y = div ~ V Consider the outward normal n = idx- jdy dx 2 + dy 2 = idx- jdy ds , and then Pdx + Qdy =-V y dx + V x dy = ( V x i + V y j ) ( idy- jdx ) = ~ V nds The by Greens theorem in the plane, Z Z A div ~ V dxdy = Z A ~ V nds Chapter7: Fourier series...
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## This note was uploaded on 07/30/2011 for the course PHZ 3113 taught by Professor Staff during the Spring '03 term at University of Central Florida.

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lec7-1 - F k = F y x- F x y So if ~ F = 0, then W = 0 as we...

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