Analysis (calculus) From the point of view of
functional analysis,
calculus is the study of two linear operators: the
differential operator \frac{\ \mathrm{d}\ }{ \mathrm{d} t }, and the
Volterra operator \int_0^t.
Fundamental analysis operators on scalar and vector fields Three operators are key to
vector calculus: • Grad (
gradient), (with operator symbol
\nabla) assigns a vector at every point in a scalar field that points in the direction of greatest rate of change of that field and whose norm measures the absolute value of that greatest rate of change. • Div (
divergence), (with operator symbol
{\nabla \cdot}) is a vector operator that measures a vector field's divergence from or convergence towards a given point. •
Curl, (with operator symbol
\nabla \!\times) is a vector operator that measures a vector field's curling (winding around, rotating around) trend about a given point. As an extension of vector calculus operators to physics, engineering and tensor spaces, grad, div and curl operators also are often associated with
tensor calculus as well as vector calculus.
Geometry In
geometry, additional structures on
vector spaces are sometimes studied. Operators that map such vector spaces to themselves
bijectively are very useful in these studies, they naturally form
groups by composition. For example, bijective operators preserving the structure of a vector space are precisely the
invertible linear operators. They form the
general linear group under composition. However, they
do not form a vector space under operator addition; since, for example, both the identity and −identity are
invertible (bijective), but their sum, 0, is not. Operators preserving the
Euclidean metric on such a space form the
isometry group, and those that fix the origin form a subgroup known as the
orthogonal group. Operators in the orthogonal group that also preserve the orientation of vector tuples form the
special orthogonal group, or the group of rotations.
Probability theory Operators are also involved in probability theory, such as
expectation,
variance, and
covariance, which are used to name both number statistics and the operators which produce them. Indeed, every covariance is basically a
dot product: Every variance is a dot product of a vector with itself, and thus is a
quadratic norm; every standard deviation is a norm (square root of the quadratic norm); the corresponding cosine to this dot product is the
Pearson correlation coefficient; expected value is basically an integral operator (used to measure weighted shapes in the space).
Fourier series and Fourier transform The Fourier transform is useful in applied mathematics, particularly physics and signal processing. It is another integral operator; it is useful mainly because it converts a function on one (temporal) domain to a function on another (frequency) domain, in a way effectively
invertible. No information is lost, as there is an inverse transform operator. In the simple case of
periodic functions, this result is based on the theorem that any continuous periodic function can be represented as the sum of a series of
sine waves and cosine waves:f(t)=\frac{\ a_0\ }{2}+\sum_{n=1}^{\infty}\ a_n\cos(\omega\ n\ t) + b_n\sin(\omega\ n\ t) The tuple is in fact an element of an infinite-dimensional vector space Sequence space|, and thus Fourier series is a linear operator. When dealing with general function \mathbb{R} \to \mathbb{C}, the transform takes on an
integral form: :f(t) = {1\over\sqrt{2\pi}}\int_{-\infty}^{+\infty}{g(\omega)\ e^{i\ \omega\ t}\ \mathrm{d}\ \omega}
Laplace transform The
Laplace transform is another integral operator and is involved in simplifying the process of solving differential equations. Given , it is defined by: F(s)=\operatorname\mathcal{L}\{f\}(s)=\int_0^\infty e^{-s\ t}\ f(t)\ \mathrm{d}\ t ==Footnotes==