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1.1 Notations and ideas
2 Operators in physics1.2 Calculus and operators 3 Operators in programming 4 See also |
Operators generally transform functions into other functions. We also say an operator maps a function to another. In some literature, they are designated by showing a small uphat 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 called Q and operates on a function f, we write Qf and not usually Q(f), however this latter notation may be used for clarity if there is a product for instance, eg. Q(fg). Throughout this article we will use Q to denote a general operator, and xi to denote the i-th argument.
Notations for operations on functions may also be notated as the following. If f(x) is a function of x and Q is the general operator we can write Q acting on f as:
Functions can be considered operators, but are generally thought of differently conceptually. "Numbers" can be considered functions too, if f(x)=x0, this represents the number 1. Similarly after multiplication by a constant, any number can be defined. When an operator takes some numbers as arguments, we can consistenly regard the operator as still transforming functions, since we have seen that numbers can be considered as functions.
Linear operators with respect to mappings between vector spaces are known more commonly as linear transformations or linear mappings.
Such an example of a linear transformation between vectors in R2 is reflection, given a vector x=(x1, x2)
Additionally, we have some other operators which we can define on functions. One such fundamental operator is that of function composition. Given two functions x1=f(t) and x2=g(t), define the operator Q:
Operators are also involved in probability theory.
Such operators as expectation, variance, covariance, factorials, et al.
Factorials are essential to the combination and permutation functions of probability and combinatorics, and are also the most commonly known postfix operator, being denoted by a ! placed after the number it expands. Its expansion follows the pattern,
Calculus is, essentially, the study of one particular operator, and its behavior embodies and exemplifies the idea of the operator in great clarity. This key operator we study in Calculus is the differential operator.
The differential operator is the symbolism used in Calculus to denote the action of taking a derivative. Common notations are such d/dx, y'(x) to denote the derivative of y(x). However here 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.
If f is a function of n variables t1,t1,...,tn, we write
The act of integration is also equivalent somewhat to taking the derivative backwards. So, in a sense it is differentiating -1 times, so we have integration in terms of the differential operator:
Given that integration is an operator as well, we have some important operators we can write in terms of integration.
The convolution of two functions is a mapping from two functions to one other, defined by an integral as follows:
If x1=f(t) and x2=g(t), define the operator Q such that;
The Fourier transform is another integral operator, and is used in many areas, not only in mathematics, but in physics and in signal processing, to name a few.
It is based on the theorem that any continuous periodic function can be represented as the sum of a series of sine waves:
f(x) = ∑ A1 sin ω + A2 sin ω/2 + A3 sin ω/3 + ...
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:
In physics, an operator often takes on a more specialized meaning that in mathematics. It often means a linear transformation from a Hilbert space to another or an element of a C* algebra. See operator (physics).
Operators are also a key part of the theory of quantum mechanics.
Programming languages, being that computers are mathematical devices, have a set of operators that perform various functions.
The arithmetic operators are the same as the mathematical ones while the bit (binary digit) operations deal with the binary number system. The logical operators determine boolean values. The string operators manipulate strings of text and there are operators which allocate segments of memory for use.
Operators are also terms for some functionality in programming languages. Consider the C programming language syntax for pointers, using the operators * and &. sizeof is sometimes considered an operator, and in C++, new and delete are also operators.
In object oriented languages, like C++, you can define your own uses for operators.
Operators in mathematics
also.Notations and ideas
Describing operators
Operators are described usually by the number of operands:
and so on.Notating operators
There are three major ways of writing operators and their arguments. These are
Q(x1, x2,...,xn).
(x1, x2,...,xn) Q
x1 Q x2Linear operators
A key concept is the concept of the linear operator. Linear operators are those which satisfy the following conditions; take the general operator Q, the function acted on under the operator Q, written as f(x), and the constant a:
Such examples of linear operators are the differential operator and Laplacian operator, which we will see later.
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.Function composition
We write this operator infix using a small circle. So, with the same definitions as before,
Probability theory
Calculus and operators
The differential operator
Notations
to represent the action of differentiating f with respect to ti.
If we differentiate f, k times, we write
How does the differential operator exemplify the idea of the operator?
Consider the function f=x2. Elementary calculus tells us that D f = 2x, futhermore if f=xα, D f = αxα-1. So we see clearly that the differential operator maps, or transforms, functions of the form xα to functions αxα-1.
It is clear that integration thus is equivalent to differentiation, so integration acts just like an operator as well -- mapping functions to functions.Integral operators
Convolution
which we write as .Fourier transform
Laplacian transform
Operators in physics
Operators in programming