ntegral 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 calculus are applied to curves, surfaces, and other geometric entities. The approach in classical differential geometry involves the use of coordinate geometry (see analytic geometry; Cartesian coordinates),
..... Click the link for more information. , in which the methods of analysis are brought to bear on geometric problems. These fields are now in a vigorous state of development.
The term applied mathematics loosely designates a wide range of studies with significant current use in the empirical sciences. It includes numerical methods and computercomputer,
device capable of performing a series of arithmetic or logical operations. A computer is distinguished from a calculating machine, such as an electronic calculator, by being able to store a computer program (so that it can repeat its operations and make logical
..... Click the link for more information. 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 tunnelswind tunnel,
apparatus for studying the interaction between a solid body and an airstream. A wind tunnel simulates the conditions of an aircraft in flight by causing a high-speed stream of air to flow past a model of the aircraft (or part of an aircraft) being tested.
..... Click the link for more information. , 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, probabilityprobability,
in mathematics, assignment of a number as a measure of the "chance" that a given event will occur. There are certain important restrictions on such a probability measure.
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