PrefaceThis book is directed to people who have a good understanding of the concepts of onevariable calculus including the notions of limit of a sequence and completeness of R. Itdevelops real analysis for functions of many real variables. It is intended to follow mybook on real analysis of functions of one variable. This is not suitable as a first course incalculus. The emphasis is on basic concepts from topology, the derivative and the integral.It does not go into functional analysis.

In order to do multivariable calculus correctly, you must first understand some linearalgebra. One cannot escape the fact that the derivative is a linear transformation, for exam-ple. Therefore, a condensed course in linear algebra is presented first, emphasizing thosetopics in linear algebra which are useful in analysis, not those topics which are primarilydependent on row operations. It is best to have had a good linear algebra course beforeattempting a book like this one, however.

I chose to feature the Lebesgue integral because I have gone through the theory ofthe Riemann integral for a function of n variables and ended up thinking it was too fussyand that the extra abstraction of the Lebesgue integral was worthwhile in order to avoidthis fussiness and to also get much better theorems. Also, it seemed to me that this bookshould be in some sense “more advanced” than my Engineering Math book which has adevelopment of the Riemann integral as an appendix.

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