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...Contents chapter multiple integrals double changing to better coordinates triple cylindrical and spherical vector calculus fields line green s theorem surface the divergence stokes curl of f mathematics after linear algebra differential equations discrete study guide for answers odd numbered problems index table this shows how integrate functions two or more variables first a integral is defined as limit sums second we find fast way compute it key idea replace by ordinary single sf x y dy dx starts with each fixed integ rate respect answer depends on now again time limits integration need care attention frequently those are hardest part why bother in place reasons there has be definition computation fall back when difficult impossible also emphasize represent than area volume words pictures that go them easiest understand you can almost see sum slices thin sticks true applications mostly other things but central add up small pieces take always same begin r graph z curved above xy plane...