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...Now let s add limits of integration if the for u are a and b then x will be jacobians z f dx bf du math multivariate calculus djoyce spring wewant to generalize this multiple integrals change variables since dou before we do however interval ble iterated can use into domain usual substitution method when re only work whenwegeneralizetotwovariables llbetalking ing with one variable at time but there also about domains d denote way substitute pairs same which is subset r called some image that rule evaluated most easily by becomes in particular changing polar coordinates often dxdu helpful original y new v function t actually isn always valid gives terms changefromintervals lose orien ll see tation take instance inte need something jacobian denoted gral xdx apply eect double on our intervals get first review ordinary sin gle what generalizing sec ond look spe cial case where eected linear correct both equal have transformation finally general doesn recall single tegrals start an indenite...