Note: in this section, "
up to" some equivalence relation E means that the canonical form is not unique in general, but that if one object has two different canonical forms, they are E-equivalent.
Large number notation Standard form is used by many mathematicians and scientists to write extremely
large numbers in a more concise and understandable way, the most prominent of which being the
scientific notation.
Number theory •
Canonical representation of a positive integer • The canonical form of a
continued fraction for representing a number is the
simple continued fraction Linear algebra Algebra Geometry In
analytic geometry: • The equation of a line:
Ax +
By =
C, with
A2 +
B2 = 1 and
C ≥ 0 • The equation of a circle: (x - h)^2 + (y - k)^2 = r^2 By contrast, there are alternative forms for writing equations. For example, the equation of a line may be written as a
linear equation in
point-slope and
slope-intercept form.
Convex polyhedra can be put into
canonical form such that: • All faces are flat, • All edges are tangent to the unit sphere, and • The centroid of the polyhedron is at the origin.
Integrable systems Every differentiable
manifold has a
cotangent bundle. That bundle can always be endowed with a certain
differential form, called the
canonical one-form. This form gives the cotangent bundle the structure of a
symplectic manifold, and allows vector fields on the manifold to be integrated by means of the
Euler-Lagrange equations, or by means of
Hamiltonian mechanics. Such systems of integrable
differential equations are called
integrable systems.
Dynamical systems The study of
dynamical systems overlaps with that of
integrable systems; there one has the idea of a
normal form (dynamical systems).
Three dimensional geometry In the study of manifolds in three dimensions, one has the
first fundamental form, the
second fundamental form and the
third fundamental form.
Functional analysis Classical logic •
Negation normal form •
Conjunctive normal form •
Disjunctive normal form •
Algebraic normal form •
Prenex normal form •
Skolem normal form •
Blake canonical form, also known as the complete sum of prime implicants, the complete sum, or the disjunctive prime form
Set theory •
Cantor normal form of an
ordinal number Game theory •
Normal form game Proof theory •
Normal form (natural deduction) Rewriting systems The symbolic manipulation of a formula from one form to another is called a "rewriting" of that formula. One can study the abstract properties of rewriting generic formulas, by studying the collection of rules by which formulas can be validly manipulated. These are the "rewriting rules"—an integral part of an
abstract rewriting system. A common question is whether it is possible to bring some generic expression to a single, common form, the normal form. If different sequences of rewrites still result in the same form, then that form can be termed a normal form, with the rewrite being called a confluent. It is not always possible to obtain a normal form.
Lambda calculus • A lambda term is in
beta normal form if no beta reduction is possible;
lambda calculus is a particular case of an abstract rewriting system. In the untyped lambda calculus, for example, the term (\lambda x.(x x) \; \lambda x.(x x)) does not have a normal form. In the typed lambda calculus, every well-formed term can be rewritten to its normal form.
Graph theory In
graph theory, a branch of mathematics, graph canonization is the problem of finding a canonical form of a given graph
G. A canonical form is a
labeled graph Canon(
G) that is
isomorphic to
G, such that every graph that is isomorphic to
G has the same canonical form as
G. Thus, from a solution to the graph canonization problem, one could also solve the problem of
graph isomorphism: to test whether two graphs
G and
H are isomorphic, compute their canonical forms Canon(
G) and Canon(
H), and test whether these two canonical forms are identical.
Computing In
computing, the reduction of data to any kind of canonical form is commonly called
data normalization. For instance,
database normalization is the process of organizing the
fields and
tables of a
relational database to minimize
redundancy and dependency. In the field of
software security, a common
vulnerability is unchecked malicious input (see
Code injection). The mitigation for this problem is proper
input validation. Before input validation is performed, the input is usually normalized by eliminating encoding (e.g.,
HTML encoding) and reducing the input data to a single common
character set. Other forms of data, typically associated with
signal processing (including
audio and
imaging) or
machine learning, can be normalized in order to provide a limited range of values. In
content management, the concept of a
single source of truth (SSOT) is applicable, just as it is in
database normalization generally and in
software development. Competent
content management systems provide logical ways of obtaining it, such as
transclusion. ==See also==