Operator Topics

• laplacian operator
• dy dx
• differential operator
• mathematical expressions
• linear transformation
• differential equations
• infix operators
• integral operator
• linear algebra
• oliver heaviside
• integral operators
• linear operators
• hilbert space
• functional analysis
• operands
• operand
• nonlinearity
• generalization
• postfix
• notations

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Operator, Operator

Often, an "operator" is a function which acts on functions to produce other functions (the sense in which Oliver Heaviside used the term); or it may be a generalization of such a function, as in linear algebra, where some of the terminology reflects the origin of the subject in operations on the functions which are solutions of differential equations.An operator can perform a function on any number of operands (inputs) though most often there is only one operand.

In analysis an "operator" may be a differential operator, to perform ordinary differentiation, or an integral operator, to perform ordinary integration.

The differential and integral operators, for example, have domain and codomains whose element are mathematical expressions of indefinite complexity. In contrast, functions with vector-valued domains but scalar ranges are called functionals and forms.

Examples include infix operators such as addition "+" and division "/" , and postfix operators such as factorial "!" . This usage is unrelated to the complexity of the entities involved.

For example, the differential operator and Laplacian operator, which we will see later.

Many other operators one encounters in mathematics are linear, and linear operators are the most easily studied (Compare with nonlinearity).

In more advanced parts of mathematics, these operators are studied as a part of functional analysis.

Common notations are dy/dx , and y'(x) to denote the derivative of y ( x ). Here, however, we will use the notation which is closest to the operator notation we have been using; that is, using Df to represent the action of taking the derivative of f.

Functions can therefore conversely be considered operators, for which we forget some of the type baggage, leaving just labels for the domain and codomain.

In that context operator often means a linear transformation from a Hilbert space to another, or (more abstractly) an element of a C*-algebra.

For example, most languages provide a '+' (addition) operator, which adds two numbers without making a function call.

For example, in C (and many derivatives such as Java), the arithmetic operators can act on any numeric data type, while functions are only allowed to act on a single explicit type. However in C++ the distinction is blurred, since Operator overloading allows operators to be defined as functions, albeit only for data types that are not built-in.

They may also include compound operators such as " += ", which increments a variable by a given value.

This is very similar to the primitive concept of an operator in a higher-level language.

Source: Wikipedia > Operator