Branches of Mathematics !
posted at Thursday, 4 April 2013 ,
05:02
Foundations
The term FOUNDATIONS is
used to refer to the formulation and analysis of the language, axioms,
and logical methods on which all of mathematics rests. The scope and
complexity of modern mathematics requires a very fine analysis of the
formal language in which meaningful mathematical statements may be
formulated and perhaps be proved true or false. Most apparent
mathematical contradictions have been shown to derive from an imprecise
and inconsistent use of language. A basic task is to furnish a set
of axioms effectively free of contradictions and at the same time rich
enough to constitute a deductive source for all of modern mathematics.
The modern axiom schemes proposed for this purpose are all couched
within the theory of sets, originated by Georg Cantor, which now constitutes a universal mathematical language.
Algebra
Historically, ALGEBRA is
the study of solutions of one or several algebraic equations, involving
the polynomial functions of one or several variables. The case where
all the polynomials have degree one (systems of linear equations) leads
to linear algebra. The case of a single equation, in which one studies
the roots of one polynomial, leads to field theory and to the so-called
Galois theory. The general case of several equations of high degree
leads to algebraic geometry, so named because the sets of solutions of
such systems are often studied by geometric methods.
Modern
algebraists have increasingly abstracted and axiomatized the structures
and patterns of argument encountered not only in the theory of
equations, but in mathematics generally. Examples of these structures
include groups (first witnessed in relation to symmetry properties of
the roots of a polynomial and now ubiquitous throughout
mathematics), rings (of which the integers, or whole numbers, constitute
a basic example), and fields (of which the rational, real, and complex
numbers are examples). Some of the concepts of modern algebra have found
their way into elementary mathematics education in the so-called new
mathematics.
Some
important abstractions recently introduced in algebra are the notions
of category and functor, which grew out of so-called homological
algebra. Arithmetic and number theory, which are concerned with special
properties of the integers—e.g., unique factorization, primes, equations
with integer coefficients (Diophantine equations), and congruences—are
also a part of algebra. Analytic number theory, however, also applies
the nonalgebraic methods of analysis to such problems.
Analysis
The essential ingredient of ANALYSIS is
the use of infinite processes, involving passage to a limit. For
example, the area of a circle may be computed as the limiting value of
the areas of inscribed regular polygons as the number of sides of the
polygons increases indefinitely. The basic branch of analysis is
the calculus. The general problem of measuring lengths, areas, volumes,
and other quantities as limits by means of approximating polygonal
figures leads to the integral calculus. The differential calculus arises
similarly from the problem of finding the tangent line to a curve at a
point. Other branches of analysis result from the application of the
concepts and methods of the calculus to various mathematical entities.
For example, vector analysis is the calculus of functions whose
variables are vectors. Here various types of derivatives and integrals
may be introduced. They lead, among other things, to the theory of
differential and integral equations, in which the unknowns are functions
rather than numbers, as in algebraic equations. Differential equations
are often the most natural way in which to express the laws governing
the behavior of various physical systems. Calculus is one of the most
powerful and supple tools of mathematics. Its applications, both in pure
mathematics and in virtually every scientific domain, are manifold.
Geometry
The shape, size, and other properties of figures and the nature of space are in the province of GEOMETRY.
Euclidean geometry is concerned with the axiomatic study of polygons,
conic sections, spheres, polyhedra, and related geometric objects in two
and three dimensions—in particular, with the relations of congruence
and of similarity between such objects. The unsuccessful attempt to
prove the “parallel postulate” from the other axioms of Euclid led in
the 19th cent. to the discovery of two different types of non-Euclidean
geometry.
The
20th cent. has seen an enormous development of topology, which is the
study of very general geometric objects, called topological spaces, with
respect to relations that are much weaker than congruence and
similarity. Other branches of geometry include algebraic geometry
and differential geometry, in which the methods of analysis are brought
to bear on geometric problems. These fields are now in a vigorous state
of development.
Applied Mathematics
The term APPLIED MATHS loosely
designates a wide range of studies with significant current use in the
empirical sciences. It includes numerical methods and computer science,
which seeks concrete solutions, sometimes approximate, to explicit
mathematical problems (e.g., differential equations, large systems of
linear equations). It has a major use in technology for modeling and
simulation. For example, the huge wind tunnels, formerly used to test
expensive prototypes of airplanes, have all but disappeared. The entire
design and testing process is now largely carried out by computer
simulation, using mathematically tailored software. It also includes
mathematical physics, which now strongly interacts with all of the
central areas of mathematics. In addition,probability theory and
mathematical statistics are often considered parts of applied
mathematics. The distinction between pure and applied mathematics is now
becoming less significant.
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