This rigorous two-part treatment advances from functions of one variable to those of several variables. Intended for students who have already completed a one-year course in elementary calculus, it defers the introduction of functions of several variables for as long as possible, and adds clarity and simplicity by avoiding a mixture of heuristic and rigorous arguments.
The first part explores functions of one variable, including numbers and sequences, continuous functions, differentiable functions, integration, and sequences and series of functions. The second part examines functions of several variables: the space of several variables and continuous functions, differentiation, multiple integrals, and line and surface integrals, concluding with a selection of related topics. Complete solutions to the problems appear at the end of the text.
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absolutely convergent assertion assume bounded function bounded interval a,b bounded set bounded variation brieﬂy called Cauchy completes the proof Compute contained continuous ﬁrst derivatives continuous function continuously differentiable COROLLARY cos(z,n curve Darboux integrable Darboux sums deﬁnition Denote diﬂerentiable divergence theorem dx dy endpoint EOREM equal equation EXAMPLE f and g ﬁnd ﬁnite number ﬁxed formula function deﬁned function f given Green’s domain Hence improper integral inequality inﬁnite integrable over a,b Let f Let f(x Let G lim f limit point line integral mean value theorem monotone increasing natural number normal domain open set partial derivatives point of G positive integer positive number power series PROBLEMS proof of Theorem Prove Theorem rational numbers real number rectangle Riemann integrable satisﬁes say that f Section sequence Show Similarly strictly monotone Suppose supremum surface Taylor’s uniformly continuous uniformly convergent vector valued function write