The term black box theory is applied to any field, philosophy and science or otherwise where some inquiry or definition is made into the relations between the appearance of something ( exterior / outside ), i. e. here specifically the things black box state, related to its characteristics and behaviour within ( interior / inner ).

Furthermore, using the definition of the exterior derivative, it can be shown that I ( D ) is closed under exterior differentiation ( it is a differential ideal ) if and only if D is involutive.

Sometimes it is convenient to extend the definition of D to arbitrary E-valued forms, thus regarding it as a differential operator on the tensor product of E with the full exterior algebra of differential forms.

By the definition of the exterior derivative, if and are arbitrary vector fields then

A simple calculation, using the definition of the exterior derivative d, yields

To relate it to the previous definition, which only involved the Maurer – Cartan form ω, take the exterior derivative of ( 1 ):

By definition of and the fact that exterior differentiation d satisfies, one has

In computational geometry, several important computational tasks involve inputs in the form of a simple polygon ; in each of these problems, the distinction between the interior and exterior is crucial in the problem definition.

By definition, the electromagnetic tensor is the exterior derivative of the differential 1-form:

The same definition holds in any unital ring or algebra where a is any invertible element.

In addition to its obvious importance in geometry and calculus, area is related to the definition of determinants in linear algebra, and is a basic property of surfaces in differential geometry.

In the Cartesian approach, the axioms are the axioms of algebra, and the equation expressing the Pythagorean theorem is then a definition of one of the terms in Euclid's axioms, which are now considered theorems.

The mathematical definition of an elementary function, or a function in elementary form, is considered in the context of differential algebra.

The notion of a homomorphism can be given a formal definition in the context of universal algebra, a field which studies ideas common to all algebraic structures.

The concrete definition given above is easy to work with, but has some minor problems: to use it we first need to represent a Lie group as a group of matrices, but not all Lie groups can be represented in this way, and it is not obvious that the Lie algebra is independent of the representation we use.

the general definition of the Lie algebra of a Lie group ( in 4 steps ):

One definition casts quasigroups as a set with one binary operation, and the other is a version from universal algebra which describes a quasigroup by using three primitive operations.

In this narrower definition, universal algebra can be seen as a special branch of model theory, in which we are typically dealing with structures having operations only ( i. e. the type can have symbols for functions but not for relations other than equality ), and in which the language used to talk about these structures uses equations only.

Now, this definition of a group is problematic from the point of view of universal algebra.

... while in the universal algebra definition there are

The definition of a Clifford algebra endows it with more structure than a " bare " K-algebra: specifically it has a designated or privileged subspace that is isomorphic to V. Such a subspace cannot in general be uniquely determined given only a K-algebra isomorphic to the Clifford algebra.

The general notion of a congruence relation can be given a formal definition in the context of universal algebra, a field which studies ideas common to all algebraic structures.

The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root ( real coefficients and roots being within the definition of complex numbers ).

such that the following properties ( modeled on the group axioms – more precisely, on the definition of a group used in universal algebra ) are satisfied

In fact, this is the definition of a Dedekind domain used in Bourbaki's " Commutative algebra ".

These types are specified by insisting on some further axioms, such as commutativity or associativity of the multiplication operation, which are not required in the broad definition of an algebra.

Each algebra consists of linear combinations of three basis elements, 1 ( the identity element ), a and b. Taking into account the definition of an identity element, it is sufficient to specify

Since the definition of the spectrum does not mention any properties of B ( X ) except those that any such algebra has, the notion of a spectrum may be generalised to this context by using the same definition verbatim.

One can extend the definition of spectrum for unbounded operators on a Banach space X, operators which are no longer elements in the Banach algebra B ( X ).

This extends the definition for bounded linear operators B ( X ) on a Banach space X, since B ( X ) is a Banach algebra.

On the other hand, the simplicity of the algebra in this proof perhaps makes it easier to understand than a proof using the definition of differentiation directly.

Its domain is the powerset of A ( with the empty set removed ), and so makes sense for any set A, whereas with the definition used elsewhere in this article, the domain of a choice function on a collection of sets is that collection, and so only makes sense for sets of sets.

Another definition of asymmetric makes asymmetry equivalent to antisymmetry plus irreflexivity.

He warns that this is open to the grave objection that it makes grace a ( quasi ) material commodity and represents an almost mechanical method of imparting what is by definition a free gift.

This view entails the problem that it makes any moral criticism of the law impossible: if conformity with natural law forms a necessary condition for legal validity, all valid law must, by definition, count as morally just.

This second definition makes sense without the axiom of choice.

It is this step that makes the definition recursive.

However, government in its broadest definition refers simply to the smaller group of people who makes and enforces decisions that affect conduct within some larger group.

The definition makes no declaration about the Church's belief that the Blessed Virgin was sinless, in the sense of actual or personal sin.

It is known that if 2 < sup > p </ sup > − 1 is prime then p is prime, so it makes no difference which Mersenne number definition is used.

The originally intended definition of the metre as 10 < sup >− 7 </ sup > of a half-meridian arc makes the mean historical nautical mile exactly ( 2 )/ = historical metres.

which makes angular acceleration directly proportional to θ, satisfying the definition of simple harmonic motion.

" Under this definition, any increase in wealth — whether through wages, benefits, bonuses, sale of stock or other property at a profit, bets won, lucky finds, awards of punitive damages in a lawsuit, qui tam actions — are all within the definition of income, unless the Congress makes a specific exemption, as it has for items such as life insurance proceeds received by reason of the death of the insured party, gifts, bequests, devises and inheritances, and certain scholarships.

The use of such a literal definition produces other problems, since the area around the church is no longer residential and the noise of the area makes it unlikely that many people would be born within earshot of the bells anymore, although The Royal London Hospital, Guy's Hospital and St Thomas ' hospital are both within the defined area covered by the sound of the Bow Bells.

It is not obvious that the first definition makes sense: why should there exist a unique minimal convex set containing X, for every X?

More formally, we can think of a truth condition as what makes for the truth of a sentence in an inductive definition of truth ( for details, see the semantic theory of truth ).

Legend is distinguished from the genre of chronicle by the fact that legends apply structures that reveal a moral definition to events, providing meaning that lifts them above the repetitions and constraints of average human lives and giving them a universality that makes them worth repeating through many generations.

The " square meters " figure of a house in Europe may report the total area of the walls enclosing the home, thus including any attached garage and non-living spaces, which makes it important to inquire what kind of surface definition has been used.

There is no formal, legal definition of what makes somebody a Co-Founder.

A circular definition crept into the classic definition of death that was once " the permanent cessation of the flow of vital bodily fluids ", which raised the question " what makes a fluid vital?

The parody is also an erroneous example of the activity of giving a definition in a dictionary, where the more general error it makes is in mistaking dictionaries to involve procedures that are found in logical or mathematical contexts.

This makes it possible to edit both standard-definition broadcast quality and high definition broadcast quality very quickly on normal PCs which do not have the power to do the full processing of the huge full-quality high-resolution data in real-time.

The ethics Siegel taught —" the art of enjoying justice "— includes this definition of good will: " The desire to have something else stronger and more beautiful, for this desire makes oneself stronger and more beautiful ".