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Math 6B NotesWritten by Victoria Kala[email protected]SH 6432u Office Hours: R 12:30-1:30pmLast updated 6/1/2016Green’s TheoremWe say that a closed curve ispositively orientedif it is oriented counterclockwise. It is negativelyoriented if it is oriented clockwise.Theorem(Green’s Theorem).LetCbe a positively oriented, piecewise smooth, simple closed curvein the plane and letDbe the region bounded byC. IfF(x, y) = (P(x, y), Q(x, y))withPandQhaving continuous partial derivatives on an open region that containsD, thenZCF·ds=ZCPdx+Qdy=ZZD@Q@x-@P@ydA.In other words, Green’s Theorem allows us to change from a complicated line integral over a curveCto a less complicated double integral over the region bounded byC.The OperatorrWe define the vector dierential operatorr(called “del”) asr=@@xi+@@yj+@@zk.Thegradientof a functionfis given byrf=@f@xi+@f@yj+@f@zk.Thecurlof a functionF=Pi+Qj+Rkis defined ascurlF=r ⇥F.Written out more explicitly:curlF=r ⇥F=ijk@@x@@y@@zPQR=@@y@@zQRi-@@x@@zPRj+@@x@@yPQk=@R@y-@Q@zi+@P@z-@R@xj+@Q@x-@P@yk
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Thedivergenceof functionF=Pi+Qj+Rkis defined asdivF=r·F=@P@x+@Q@y+@R@z.Notice that the curl of a function is a vector and the divergence of a function is a scalar (just anumber).Surface IntegralsIfFis a continuous vector field defined on an oriented surfaceSwith unit normal vectorn, thenthe surface integral ofFoverSisZZSF·dS=ZZSF·ndS.IfSis given by a vector functionr(u, v), then the above integral can be written asZZSF·dS=ZZDF·(rurv)dAwhereDis the parameter domain (the projection ofSonto a plane).If we are givenSis the functionz=g(x, y), we can define the parametrization ofSto ber(x, y) = (x, y, g(x, y)).Then the surface integral ofFoverSis given byZZSF·dS=ZZDF·(rxry)dAwhereDis the projection ofz=g(x, y) onto thexyplane.Stokes’ TheoremTheorem(Stokes’ Theorem).LetSbe an oriented piecewise-smooth surface that is bounded bya simple, closed, piecewise-smooth boundary curveCwith positive orientation. LetFbe a vectorfield whose components have continuous partial derivatives on an open region inR3that containsS. ThenZCF·dr=ZZScurlF·dS.In other words, the line integral around the boundary curve of a surfaceSof the tangential com-ponent ofFis equal to the surface integral of the normal component of the curl ofF.Note: This looks very similar to Green’s Theorem! In fact, Green’s Theorem is a special case ofStokes’ Theorem; it is whenFis restricted to thexyplane. Stokes’ Theorem can then be thoughtof as the higher-dimensional version of Green’s Theorem.
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Divergence TheoremTheorem(Divergence Theorem).LetEbe a simple solid region and letSbe the boundary surfaceofE, given with positive (outward) orientation. LetFbe a vector field whose component functionshave continuous partial derivatives on an open region that containsE. ThenZZSF·dS=ZZZEdivFdV.In other words, the divergence theorem states that the flux ofFacross the boundary surface ofE

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