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Exterior Covariant Derivative
In the mathematical field of differential geometry, the exterior covariant derivative is an extension of the notion of exterior derivative to the setting of a differentiable principal bundle or vector bundle with a connection. Definition Let ''G'' be a Lie group and be a principal ''G''-bundle on a smooth manifold ''M''. Suppose there is a connection on ''P''; this yields a natural direct sum decomposition T_u P = H_u \oplus V_u of each tangent space into the horizontal and vertical subspaces. Let h: T_u P \to H_u be the projection to the horizontal subspace. If ''ϕ'' is a ''k''-form on ''P'' with values in a vector space ''V'', then its exterior covariant derivative ''Dϕ'' is a form defined by :D\phi(v_0, v_1,\dots, v_k)= d \phi(h v_0 ,h v_1,\dots, h v_k) where ''v''''i'' are tangent vectors to ''P'' at ''u''. Suppose that is a representation of ''G'' on a vector space ''V''. If ''ϕ'' is equivariant in the sense that :R_g^* \phi = \rho(g)^\phi where R_g(u) = ug, then ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Vector Field
In the study of mathematics and especially differential geometry, fundamental vector fields are an instrument that describes the infinitesimal behaviour of a smooth Lie group action on a smooth manifold. Such vector fields find important applications in the study of Lie theory, symplectic geometry, and the study of Hamiltonian group actions. Motivation Important to applications in mathematics and physics is the notion of a flow on a manifold. In particular, if M is a smooth manifold and X is a smooth vector field, one is interested in finding integral curves to X . More precisely, given p \in M one is interested in curves \gamma_p: \mathbb R \to M such that : \gamma_p'(t) = X_, \qquad \gamma_p(0) = p, for which local solutions are guaranteed by the Existence and Uniqueness Theorem of Ordinary Differential Equations. If X is furthermore a complete vector field, then the flow of X , defined as the collection of all integral curves for X , is a diffeomorphism of M. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Curvature Form
In differential geometry, the curvature form describes curvature of a connection on a principal bundle. The Riemann curvature tensor in Riemannian geometry can be considered as a special case. Definition Let ''G'' be a Lie group with Lie algebra \mathfrak g, and ''P'' → ''B'' be a principal ''G''-bundle. Let ω be an Ehresmann connection on ''P'' (which is a \mathfrak g-valued one-form on ''P''). Then the curvature form is the \mathfrak g-valued 2-form on ''P'' defined by :\Omega=d\omega + omega \wedge \omega= D \omega. (In another convention, 1/2 does not appear.) Here d stands for exterior derivative, cdot \wedge \cdot/math> is defined in the article "Lie algebra-valued form" and ''D'' denotes the exterior covariant derivative. In other terms, :\,\Omega(X, Y)= d\omega(X,Y) + omega(X),\omega(Y)/math> where ''X'', ''Y'' are tangent vectors to ''P''. There is also another expression for Ω: if ''X'', ''Y'' are horizontal vector fields on ''P'', thenProof: \sigma\Omega(X ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector-valued Differential Forms
In mathematics, a vector-valued differential form on a manifold ''M'' is a differential form on ''M'' with values in a vector space ''V''. More generally, it is a differential form with values in some vector bundle ''E'' over ''M''. Ordinary differential forms can be viewed as R-valued differential forms. An important case of vector-valued differential forms are Lie algebra-valued forms. (A connection form is an example of such a form.) Definition Let ''M'' be a smooth manifold and ''E'' → ''M'' be a smooth vector bundle over ''M''. We denote the space of smooth sections of a bundle ''E'' by Γ(''E''). An ''E''-valued differential form of degree ''p'' is a smooth section of the tensor product bundle of ''E'' with Λ''p''(''T'' ∗''M''), the ''p''-th exterior power of the cotangent bundle of ''M''. The space of such forms is denoted by :\Omega^p(M,E) = \Gamma(E\otimes\Lambda^pT^*M). Because Γ is a strong monoidal functor, this can also be interpreted as :\Gamma(E\otimes\Lambda^ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Associated Bundle
In mathematics, the theory of fiber bundles with a structure group G (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from F_1 to F_2, which are both topological spaces with a group action of G. For a fiber bundle ''F'' with structure group ''G'', the transition functions of the fiber (i.e., the cocycle) in an overlap of two coordinate systems ''U''α and ''U''β are given as a ''G''-valued function ''g''αβ on ''U''α∩''U''β. One may then construct a fiber bundle ''F''′ as a new fiber bundle having the same transition functions, but possibly a different fiber. An example A simple case comes with the Möbius strip, for which G is the cyclic group of order 2, \mathbb_2. We can take as F any of: the real number line \mathbb, the interval 1,\ 1/math>, the real number line less the point 0, or the two-point set \. The action of G on these (the non-identity element acting as x\ \rightarrow\ -x in each ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Matrix
In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that :\mathbf = \mathbf = \mathbf_n \ where denotes the -by- identity matrix and the multiplication used is ordinary matrix multiplication. If this is the case, then the matrix is uniquely determined by , and is called the (multiplicative) ''inverse'' of , denoted by . Matrix inversion is the process of finding the matrix that satisfies the prior equation for a given invertible matrix . A square matrix that is ''not'' invertible is called singular or degenerate. A square matrix is singular if and only if its determinant is zero. Singular matrices are rare in the sense that if a square matrix's entries are randomly selected from any finite region on the number line or complex plane, the probability that the matrix is singular is 0, that is, it will "almost never" be singular. Non-square matrices (-by- matrices for which ) do not hav ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Bracket Of Vector Fields
In the mathematical field of differential topology, the Lie bracket of vector fields, also known as the Jacobi–Lie bracket or the commutator of vector fields, is an operator that assigns to any two vector fields ''X'' and ''Y'' on a smooth manifold ''M'' a third vector field denoted . Conceptually, the Lie bracket is the derivative of ''Y'' along the flow generated by ''X'', and is sometimes denoted ''\mathcal_X Y'' ("Lie derivative of Y along X"). This generalizes to the Lie derivative of any tensor field along the flow generated by ''X''. The Lie bracket is an R- bilinear operation and turns the set of all smooth vector fields on the manifold ''M'' into an (infinite-dimensional) Lie algebra. The Lie bracket plays an important role in differential geometry and differential topology, for instance in the Frobenius integrability theorem, and is also fundamental in the geometric theory of nonlinear control systems., nonholonomic systems; , feedback linearization. Definitio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector-valued Differential Forms
In mathematics, a vector-valued differential form on a manifold ''M'' is a differential form on ''M'' with values in a vector space ''V''. More generally, it is a differential form with values in some vector bundle ''E'' over ''M''. Ordinary differential forms can be viewed as R-valued differential forms. An important case of vector-valued differential forms are Lie algebra-valued forms. (A connection form is an example of such a form.) Definition Let ''M'' be a smooth manifold and ''E'' → ''M'' be a smooth vector bundle over ''M''. We denote the space of smooth sections of a bundle ''E'' by Γ(''E''). An ''E''-valued differential form of degree ''p'' is a smooth section of the tensor product bundle of ''E'' with Λ''p''(''T'' ∗''M''), the ''p''-th exterior power of the cotangent bundle of ''M''. The space of such forms is denoted by :\Omega^p(M,E) = \Gamma(E\otimes\Lambda^pT^*M). Because Γ is a strong monoidal functor, this can also be interpreted as :\Gamma(E\otimes\Lambda^ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connection (vector Bundle)
In mathematics, and especially differential geometry and gauge theory, a connection on a fiber bundle is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. The most common case is that of a linear connection on a vector bundle, for which the notion of parallel transport must be linear. A linear connection is equivalently specified by a '' covariant derivative'', an operator that differentiates sections of the bundle along tangent directions in the base manifold, in such a way that parallel sections have derivative zero. Linear connections generalize, to arbitrary vector bundles, the Levi-Civita connection on the tangent bundle of a pseudo-Riemannian manifold, which gives a standard way to differentiate vector fields. Nonlinear connections generalize this concept to bundles whose fibers are not necessarily linear. Linear connections are also called Koszul connections after Jean-Louis Koszul, who g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flat Connection
In differential geometry, the curvature form describes curvature of a connection on a principal bundle. The Riemann curvature tensor in Riemannian geometry can be considered as a special case. Definition Let ''G'' be a Lie group with Lie algebra \mathfrak g, and ''P'' → ''B'' be a principal ''G''-bundle. Let ω be an Ehresmann connection on ''P'' (which is a \mathfrak g-valued one-form on ''P''). Then the curvature form is the \mathfrak g-valued 2-form on ''P'' defined by :\Omega=d\omega + omega \wedge \omega= D \omega. (In another convention, 1/2 does not appear.) Here d stands for exterior derivative, cdot \wedge \cdot/math> is defined in the article "Lie algebra-valued form" and ''D'' denotes the exterior covariant derivative. In other terms, :\,\Omega(X, Y)= d\omega(X,Y) + omega(X),\omega(Y)/math> where ''X'', ''Y'' are tangent vectors to ''P''. There is also another expression for Ω: if ''X'', ''Y'' are horizontal vector fields on ''P'', thenProof: \sigma\Omega(X ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of atoms and molecules. Electromagnetism can be thought of as a combination of electricity and magnetism, two distinct but closely intertwined phenomena. In essence, electric forces occur between any two charged particles, causing an attraction between particles with opposite charges and repulsion between particles with the same charge, while magnetism is an interaction that occurs exclusively between ''moving'' charged particles. These two effects combine to create electromagnetic fields in the vicinity of charge particles, which can exert influence on other particles via the Lorentz force. At high energy, the weak force and electromagnetic force are unified as a single electroweak force. The electromagnetic force is responsible for many o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field Strength Tensor
In electromagnetism, the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field in spacetime. The field tensor was first used after the four-dimensional tensor formulation of special relativity was introduced by Hermann Minkowski. The tensor allows related physical laws to be written very concisely, and allows for the quantization of the electromagnetic field by Lagrangian formulation described below. Definition The electromagnetic tensor, conventionally labelled ''F'', is defined as the exterior derivative of the electromagnetic four-potential, ''A'', a differential 1-form: :F \ \stackrel\ \mathrmA. Therefore, ''F'' is a differential 2-form—that is, an antisymmetric rank-2 tensor field—on Minkowski space. In component form, :F_ = \partial_\mu A_\nu - \partial_\nu A_\mu. where \partial is the four-gradient and A is the fou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
