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...Example evaluate the surface integral of vector eld f x i yxj k over s that is graph z y rectangle solution use formula for a g zz ds d xi yj dxdy in our case we get yx dy ydy let be triangle with vertices and xyz j calculate if oriented by downward normal since lies plane see right hand part figure it linear function ax c v u substituting equation b which can solve to nd e may take as parameters or domain parametrization at this t third component positive orientation determined upward so will have multiply answer now minus sign reminding us wrong zzsf zzdxyz dudv xyzdudv uv compute double draw integration left reduction iterated integrals dvdu carrying out du multiplying simplifying becomes note related scalar homework set exercise section asks you xyzds where same perhaps make some calculations from n obtained problem equations describe disk radius lying suppose r position zk parallel xy outward unit hence thus nds zds alternatively using graphs gi gj zzsr zzd area ofd...