ANALYSIS BASICS
What is mathematical analysis?
Mathmatical analysis is a branch of mathematics that uses the concepts and methods
common to the field of calculus. The key to mathematical analysis is the use of infinite
processes; in turn, that involves passage to a limit, or, in other words, the basic branch
of calculus. For example, a circle’s area can be thought of as the limiting value of the
areas of regular polygons within the circle, as the number of the polygons’ sides
increase indefinitely. The reason for calculus is simple: It is one of the most powerful
and flexible tools not only in mathematics but in virtually every scientific field.
What are the various forms of calculus?
As mentioned above, calculus uses infinite processes that involve passage to a limit. In
order to solve its functions, it includes a formal set of mathematical rules applied to
changing quantities. Many mathematicians break down calculus into two branches.
The first is integral calculus,or the general problems of measuring length, area, vol-
ume, and other quantities as limits by approximating polygonal shapes; it finds the
quantity when the rate of change is known. The second branch deals with discovering
the tangent line to a curve at a specific point—or differential calculus. In this case,
the calculus determines the quantity’s rate of change. There also are several other
forms of mathematical analysis that depend on calculus methods, such as vector
analysis, tensor analysis, differential geometry, and complex variable analysis.
What culture took the first steps in the development of mathematical analysis?
Mathematical analysis—and, thus, the ideas of calculus—took centuries to develop.
Probably the first to present some solid concepts in the field were the Greeks whose 209
MATHEMATICAL
ANALYSIS