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Lecture

Definitions and Notations

Description

This lecture covers the definitions and notations related to functions, including the domain of definition, image of a function, and the concepts of surjective and injective functions.

In MOOCs (2)

Analysis I (part 1): Prelude, basic concepts, real numbers

Analysis I

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Related concepts (32)

Jacobi elliptic functions

In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions. They are found in the description of the motion of a pendulum (see also pendulum (mathematics)), as well as in the design of electronic elliptic filters. While trigonometric functions are defined with reference to a circle, the Jacobi elliptic functions are a generalization which refer to other conic sections, the ellipse in particular. The relation to trigonometric functions is contained in the notation, for example, by the matching notation for .

Trigonometric functions

In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis.

Injective module

In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers. Specifically, if Q is a submodule of some other module, then it is already a direct summand of that module; also, given a submodule of a module Y, any module homomorphism from this submodule to Q can be extended to a homomorphism from all of Y to Q. This concept is to that of projective modules.

Lemniscate elliptic functions

In mathematics, the lemniscate elliptic functions are elliptic functions related to the arc length of the lemniscate of Bernoulli. They were first studied by Giulio Fagnano in 1718 and later by Leonhard Euler and Carl Friedrich Gauss, among others. The lemniscate sine and lemniscate cosine functions, usually written with the symbols sl and cl (sometimes the symbols sinlem and coslem or sin lemn and cos lemn are used instead), are analogous to the trigonometric functions sine and cosine.

Injective hull

In mathematics, particularly in algebra, the injective hull (or injective envelope) of a module is both the smallest injective module containing it and the largest essential extension of it. Injective hulls were first described in . A module E is called the injective hull of a module M, if E is an essential extension of M, and E is injective. Here, the base ring is a ring with unity, though possibly non-commutative. An injective module is its own injective hull. The injective hull of an integral domain is its field of fractions .

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