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...Solutions of some homework problems for math p let c consist straight lines joining and s be the triangular surface with these vertices verify stokes theorem directly f yzi xzj xyk solution sneaky easy way note that xyz so curl rr ds begins ends at same r point say cf find area disk d radius using green y dx x dy identity mapping which is denitely a parameterization i dxdy by this equal to since t hrcost rsinti parameterizes boundary curve h rsint rcosti nout hcost sinti outward pointing we have z signdet cost sint rcost signr rsintdx rcostdy dt sin cos tdt divergence show nds wheref yi xjandd unit b as ver sion in problem ndsis classical notation ux through e checked sign says ydx xdy ft ydy xdx therefore costdx sintdy costsint sintcost region holds suppose harmonic on prove parameterize then q assuming investigate why fails they mean form qdy needs open subset u containing closed order it least function from into if were would bounded near but not example thus even continuous hence d...