ysisanalysis,
branch of mathematics that utilizes the concepts and methods of the calculus. It includes not only basic calculus, but also advanced calculus, in which such underlying concepts as that of a limit are subjected to rigorous examination; differential and integral
..... Click the link for more information. is the use of infinite processes, involving passage to a limitlimit,
in mathematics, value approached by a sequence or a function as the index or independent variable approaches some value, possibly infinity. For example, the terms of the sequence 1-2, 1-4, 1-8, 1-16, … are obviously getting smaller and smaller; since, if enough
..... Click the link for more information. . 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 calculuscalculus,
branch of mathematics that studies continuously changing quantities. The calculus is characterized by the use of infinite processes, involving passage to a limit—the notion of tending toward, or approaching, an ultimate value.
..... Click the link for more information. . 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, vectorvector,
quantity having both magnitude and direction; it may be represented by a directed line segment. Many physical quantities are vectors, e.g., force, velocity, and momentum.
..... Click the link for more information. 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.
The shape, size, and other properties of figures and the nature of space are in the province of geometry. Euclidean geometrygeometry
[Gr.,=earth measuring], branch of mathematics concerned with the properties of and relationships between points, lines, planes, and figures and with generalizations of these concepts.
..... Click the link for more information. 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 geometrynon-Euclidean geometry,
branch of geometry in which the fifth postulate of Euclidean geometry, which allows one and only one line parallel to a given line through a given external point, is replaced by one of two alternative postulates.
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The 20th cent. has seen an enormous development of topologytopology,
branch of mathematics, formerly known as analysis situs, that studies patterns of geometric figures involving position and relative position without regard to size.
..... Click the link for more information. , 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 geometrydifferential geometry,
branch of geometry in which the concepts of the