Information AboutOperators |
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In Mathematics , an operator is a Function that performs some sort of operation on a number, variable, or function. Addition and multiplication are simple examples of operators. Another example is the D Operator , when placed before a differentiable function ''f''(''t''), indicates that the function is to be Differentiate d with respect to the variable ''t''. Most operators perform a function on two inputs called Operands . Although this is the case, an operator can perform a function on any number of inputs - though more than two inputs can make for a confusing operator. OPERATORS AND LEVELS OF ABSTRACTION To begin with, the usage of ''operator'' in mathematics is subsumed in the usage of '' Function '': an operator can be taken to be some special kind of function. The word is generally used to call attention to some aspect of its nature as a function. Since there are several such aspects that are of interest, there is no completely consistent terminology. Common are these:
A single operator might conceivably qualify under all three of these. Other important ideas are:
These are abstract ideas from mathematics, and Computer Science . They may however also be encountered in Quantum Mechanics . There Dirac drew a clear distinction between Q-number or operator quantities, and C-number s which are conventional Complex Number s. The manipulation of ''q''-numbers from that point on became basic to theoretical physics. DESCRIBING OPERATORS Operators are described usually by the number of operands:
The number of operands is also called the ''arity'' of the operator. If an operator has an arity given as ''n''-ary (or ''n''-adic), then it takes ''n'' arguments. In programming, other than Functional Programming , the -ary terms are more often used than the other variants. See Arity for an extensive list of the -ary endings. The field of Universal Algebra also includes the study of operators and their arities. Notations There are five major ''systematic'' ways of writing operators and their arguments. These are
::''Q''(''x''1, ''x''2,...,''x''''n''). : In Prefix Notation , the brackets are sometimes omitted if it is known that ''Q'' is an ''n''-ary operator.
::(''x''1, ''x''2,...,''x''''n'') ''Q'' : In Postfix Notation , the brackets are sometimes omitted if it is known that ''Q'' is an ''n''-ary operator.
:: ''x''1 ''Q'' ''x''2.
There are other notations commonly met. Writing Exponent s such as 28 is really a law unto itself, since it is postfix only as a unary operator applied to 2, but on a slant as binary operator. In some literature, a Circumflex is written over the operator name. In certain circumstances, they are written ''unlike'' functions, when an operator has a single Argument or ''operand''. For example, if the operator name is ''Q'' and the operand a function ''f'', we write ''Qf'' and not usually ''Q''(''f''); this latter notation may however be used for clarity if there is a product — for instance, ''Q''(''fg''). Later on we will use ''Q'' to denote a general operator, and ''x''''i'' to denote the ''i''-th argument. Notations for operators include the following. If ''f''(''x'') is a function of ''x'' and ''Q'' is the general operator we can write ''Q'' acting on ''f'' as (''Qf'')(''x'') also. Operators are often written in Calligraphy to differentiate them from standard functions. For instance, the Fourier Transform (an operator on functions) of ''f''(''t'') (a function of ''t''), which produces another function ''F''(ω) (a function of ω), would be represented as EXAMPLES OF MATHEMATICAL OPERATORS This section concentrates on illustrating the expressive power of the operator concept in mathematics. Please refer to individual topics pages for further details. Linear operators ''Main article'': Linear Transformation The most common kind of operator encountered are ''linear operators''. In talking about linear operators, the operator is signified generally by the letters ''T'' or ''L''. Linear operators are those which satisfy the following conditions; take the general operator ''T'', the function acted on under the operator ''T'', written as ''f''(''x''), and the constant a: : : Many operators are linear. For example, the differential operator and Laplacian operator, which we will see later. Linear operators are also known as Linear Transformation s or linear mappings. Many other operators one encounters in mathematics are linear, and linear operators are the most easily studied (Compare with Nonlinearity ). Such an example of a linear transformation between vectors in R2 is reflection: given a vector '''x''' = (''x''1, ''x''2) Q We can also make sense of linear operators between generalisations of finite- Dimensional vector spaces. For example, there is a large body of work dealing with linear operators on Hilbert Spaces and on Banach Spaces . See also Operator Algebra . Operators in probability theory ''Main article'': Probability Theory Operators are also involved in probability theory. Such operators as Expectation , Variance , Covariance , Factorial s, etc. Operators in calculus ''D'' = d/d''t'', and the Indefinite Integral Operator . These operators are ''linear'', as are many of the operators constructed from them. In more advanced parts of mathematics, these operators are studied as a part of Functional Analysis . The differential operator ''Main article'': Differential Operator The Differential Operator is an operator which is fundamentally used in Calculus to denote the action of taking a derivative. Common notations are d/d''x'', and ''y'' ′(''x'') to denote the derivative of ''y''(''x''). Here, however, we will use the notation that is closest to the operator notation we have been using; that is, using D ''f'' to represent the action of taking the derivative of ''f''. Integral operators Given that integration is an operator as well (inverse of differentiation), we have some important operators we can write in terms of integration. = Convolution ''Main article'': Convolution
= Fourier transform ''Main article'': Fourier Transform The Fourier transform is used in many areas, not only in mathematics, but in physics and in signal processing, to name a few. It is another integral operator; it is useful mainly because it converts a function on one (spatial) domain to a function on another (frequency) domain, in a way that is effectively Invertible . Nothing significant is lost, because there is an inverse transform operator. In the simple case of Periodic Function s, this result is based on the theorem that any continuous periodic function can be represented as the sum of a series of Sine Wave s and cosine waves: : When dealing with general function R → '''C''', the transform takes on an Integral form: : = Laplacian transform ''Main article:'' Laplace Transform The ''Laplace transform'' is another integral operator and is involved in simplifying the process of solving differential equations. Given ''f'' = ''f''(''s''), it is defined by: : Heading Fundamental operators on scalar and vector fields ''Main articles:'' Vector Calculus , Scalar Field , Gradient , Divergence , and Curl Three main operators are key to Vector Calculus , the operator ∇, known as Gradient , where at a certain point in a scalar field forms a vector which points in the direction of greatest change of that scalar field. In a vector field, the Divergence is an operator that measures a vector field's tendency to originate from or converge upon a given point. Curl , in a vector field, is a vector operator that shows a vector field's tendency to rotate about a point. RELATION TO TYPE THEORY ''Main article:'' Type Theory In Type Theory , an operator itself is a function, but has an attached ''type'' indicating the correct operand, and the kind of function returned. Functions can therefore conversely be considered operators, for which we forget some of the type baggage, leaving just labels for the domain and codomain. OPERATORS IN PHYSICS ''Main article:'' Operator (physics)
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