Calculus and Real analysis
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Differentiation, integration, series, and so on are familiar to students of
elementary calculus. But these topics lead in a number of distinct directions
when pursued with greater care and in greater detail. The central location
of these fields in the MathMap is indicative of the utility in other
branches of mathematics, particularly throughout analysis.
26: Real functions are those studied in
calculus classes; the focus here is on their derivatives and
integrals, and general inequalities. This category includes familiar
functions such as rational functions. This seems the most appropriate
area to receive questions concerning elementary calculus.
28: Measure theory and integration is the study
of lengths, surface area, and volumes in general spaces. This is a
critical feature of a full development of integration theory; moreover,
it provides the basic framework for probability theory. Measure theory
is a meeting place between the tame applicability of real functions
and the wild possibilities of set theory. This is the setting for fractals.
33: Special functions are just that: specialized
functions beyond the familiar trigonometric or exponential functions.
The ones studied (hypergeometric functions, orthogonal polynomials, and
so on) arise very naturally in areas of analysis, number theory,
Lie groups, and combinatorics. Very detailed information is often available.
39: Finite differences and functional
equations both involve deducing properties of functions, as in
differential equations, but the premises are different: with
difference equations, the defining relation is not a differential
equation but a difference of values of the function. Functional
equations have as premises (usually) algebraic relationships among the
values of a function at several points.
40: Sequences and series are really just the most
common examples of limiting processes; convergence criteria and rates of
convergence are as important as finding "the answer". (In the case of
sequences of functions, it's also important to find "the question"!)
One studies particular series of interest (e.g. Taylor series of known
functions), as well as general methods for computing sums rapidly, or
formally. Series can be estimated with integrals, their stability can be
investigated with analysis. Manipulations of series (e.g. multiplying
or inverting) are also of importance.
You might want to continue the tour with a trip through complex analysis.
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Last modified 2000/01/25 by Dave Rusin. Mail: