If T is a ( p, q )- tensor ( p for the contravariant vector and q for the covariant one ), then we define the divergence of T to be the ( p, q − 1 )- tensor

The standard way to do this, as carried out in the remainder of this article, is to define the Euclidean plane as a two-dimensional real vector space equipped with an inner product.

This product allows us to define the " length " of a vector x as

We could also define a Lie algebra structure on T < sub > e </ sub > using right invariant vector fields instead of left invariant vector fields.

Instead one needs to define Lie groups modeled on more general locally convex topological vector spaces.

In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a " coordinate system " ( as long as the basis is given a definite order ).

Similarly, for any semi-normed vector space we can define the distance between two vectors u and v as ‖ u − v ‖.

Once tangent spaces have been introduced, one can define vector fields, which are abstractions of the velocity field of particles moving on a manifold.

Such a vector field serves to define a generalized ordinary differential equation on a manifold: a solution to such a differential equation is a differentiable curve on the manifold whose derivative at any point is equal to the tangent vector attached to that point by the vector field.

These derivations form a real vector space if we define addition and scalar multiplication for derivations by

Specifically, if v is a tangent vector of M at a point x ( thought of as a derivation ) then define the directional derivative in the direction v by

When the length of vectors is defined, it is possible to also define a dot product — a scalar-valued product of two vectors — which gives a convenient algebraic characterization of both length ( the square root of the dot product of a vector by itself ) and angle ( a function of the dot product between any two vectors ).

However, it is not always possible or desirable to define the length of a vector in a natural way.

Metafont is a description language used to define vector fonts.

On the category of finite-dimensional vector spaces and linear maps, one can define an infranatural isomorphism from vector spaces to their dual by choosing an isomorphism for each space ( say, by choosing a basis for every vector space and taking the corresponding isomorphism ), but this will not define a natural transformation.

But first, we must define two terms so that their meaning will be clearly understood: form -- any unique sequence of alphabetic characters that can appear in a language preceded and followed by a space ; ;

To define angles in an abstract real inner product space, we replace the Euclidean dot product ( · ) by the inner product, i. e.

If X is a Banach space and K is the underlying field ( either the real or the complex numbers ), then K is itself a Banach space ( using the absolute value as norm ) and we can define the continuous dual space as X ′ = B ( X, K ), the space of continuous linear maps into K.

The continuous dual space can be used to define a new topology on X: the weak topology.

The real line R with its usual topology is a locally compact Hausdorff space, hence we can define a Borel measure on it.

Starting from any ket in this Hilbert space, we can define a complex scalar function of r, known as a wavefunction:

Four points P < sub > 0 </ sub >, P < sub > 1 </ sub >, P < sub > 2 </ sub > and P < sub > 3 </ sub > in the plane or in higher-dimensional space define a cubic Bézier curve.

This formulation is analogous to the construction of the cotangent space to define the Zariski tangent space in algebraic geometry.

If S is an arbitrary set, then the set S < sup > N </ sup > of all sequences in S becomes a complete metric space if we define the distance between the sequences ( x < sub > n </ sub >) and ( y < sub > n </ sub >) to be, where N is the smallest index for which x < sub > N </ sub > is distinct from y < sub > N </ sub >, or 0 if there is no such index.

In modern mathematics, it is more common to define Euclidean space using Cartesian coordinates and the ideas of analytic geometry.

Due to the different designs of the spacecraft, the American and Soviet space programs define the duration of an EVA differently.

To every topological space X with distinguished point x < sub > 0 </ sub >, one can define the fundamental group based at x < sub > 0 </ sub >, denoted π < sub > 1 </ sub >( X, x < sub > 0 </ sub >).

Conversely, one might expect that inertial motions, once identified by observing the actual motions of bodies and making allowances for the external forces ( such as electromagnetism or friction ), can be used to define the geometry of space, as well as a time coordinate.

He used étale coverings to define an algebraic analogue of the fundamental group of a topological space.

Finite picture whose dimensions are a certain amount of space and a certain amount of time ; the protons and electrons are the streaks of paint which define the picture against its space-time background.

The highly-crafted pristine white structural frame and all-glass walls define a simple rectilinear interior space, allowing nature and light to envelop the interior space.

In general terms, a calculus is a formal system that consists of a set of syntactic expressions ( well-formed formulæ or wffs ), a distinguished subset of these expressions ( axioms ), plus a set of formal rules that define a specific binary relation, intended to be interpreted as logical equivalence, on the space of expressions.

In integration theory, specifying a measure allows one to define integrals on spaces more general than subsets of Euclidean space ; moreover, the integral with respect to the Lebesgue measure on Euclidean spaces is more general and has a richer theory than its predecessor, the Riemann integral.

A binary operation is a binary function where the sets X, Y, and Z are all equal ; binary operations are often used to define algebraic structures.

Practically only a very small percentage of addresses is kept as initial reference points ( which also requires storage ), and most of the database data is accessed by indirection using displacement calculations ( distance in bits from the reference points ) and data structures which define access paths ( using pointers ) to all needed data in effective manner, optimized for the needed data access operations.

Practically only a very small percentage of addresses is kept as initial reference points ( which also requires storage ), and most of the database data is accessed by indirection using displacement calculations ( distance in bits from the reference points ) and data structures which define access paths ( using pointers ) to all needed data in effective manner, optimized for the needed data access operations.

The most common way to formalize this is by defining a field as a set together with two operations, usually called addition and multiplication, and denoted by + and ·, respectively, such that the following axioms hold ; subtraction and division are defined implicitly in terms of the inverse operations of addition and multiplication :< ref group =" note "> That is, the axiom for addition only assumes a binary operation The axiom of inverse allows one to define a unary operation that sends an element to its negative ( its additive inverse ); this is not taken as given, but is implicitly defined in terms of addition as " is the unique b such that ", " implicitly " because it is defined in terms of solving an equation — and one then defines the binary operation of subtraction, also denoted by "−", as in terms of addition and additive inverse.

Functional theories of grammar differ from formal theories of grammar, in that the latter seeks to define the different elements of language and describe the way they relate to each other as systems of formal rules or operations, whereas the former defines the functions performed by language and then relates these functions to the linguistic elements that carry them out.

Then and become partially defined operations on G, and will in fact be defined everywhere ; so we define * to be and to be.

* Extended Operation — generic operation used to define other operations

On the other hand, LDAP does not define transactions of multiple operations: If you read an entry and then modify it, another client may have updated the entry in the meantime.

The Extended Operation is a generic LDAP operation that can define new operations that were not part of the original protocol specification.

Examples: One can define new operations.

The purpose of polymorphism is to implement a style of programming called message-passing in the literature, in which objects of various types define a common interface of operations for users.

To obtain the expected answer of 23, parentheses must be used to explicitly define the order of operations:

For a congruence on a ring, the equivalence class containing 0 is always a two-sided ideal, and the two operations on the set of equivalence classes define the corresponding quotient ring.

One might use definitions that rely on operations in order to avoid the troubles associated with attempting to define things in terms of some intrinsic essence.

The special theory of relativity can be viewed as the introduction of operational definitions for simultaneity of events and of distance, that is, as providing the operations needed to define these terms.

The Brahmasphutasiddhanta of Brahmagupta ( 598 – 668 ) is the earliest known text to treat zero as a number in its own right and to define operations involving zero.

It is generally regarded among mathematicians that a natural way to interpret division by zero is to first define division in terms of other arithmetic operations.

Although division by zero cannot be sensibly defined with real numbers and integers, it is possible to consistently define it, or similar operations, in other mathematical structures.

One may define the operations of the algebra of sets:

Under Desargues ' Theorem, combined with the other axioms, it is possible to define the basic operations of arithmetic, geometrically.

Note that modern set-theoretic definitions usually define operations as maps between sets, so adding closure to a structure as an axiom is superfluous, though it still makes sense to ask whether subsets are closed.

This does indeed define an equivalence relation, it is compatible with the operations defined above, and the set R of all equivalence classes can be shown to satisfy all the usual axioms of the real numbers.