|
Solder Form
In mathematics, more precisely in differential geometry, a soldering (or sometimes solder form) of a fiber bundle to a smooth manifold is a manner of attaching the fibers to the manifold in such a way that they can be regarded as tangent. Intuitively, soldering expresses in abstract terms the idea that a manifold may have a point of contact (mathematics), contact with a certain model Klein geometry at each point. In extrinsic differential geometry, the soldering is simply expressed by the tangency of the model space to the manifold. In intrinsic geometry, other techniques are needed to express it. Soldering was introduced in this general form by Charles Ehresmann in 1950. Soldering of a fibre bundle Let ''M'' be a smooth manifold, and ''G'' a Lie group, and let ''E'' be a smooth fibre bundle over ''M'' with structure group ''G''. Suppose that ''G'' Group action (mathematics), acts transitively on the typical fibre ''F'' of ''E'', and that dim ''F'' = dim ''M''. A soldering of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solder Form Of A Circle Over A Torus
Solder (; North American English, NA: ) is a fusible alloy, fusible metal alloy used to create a permanent bond between metal workpieces. Solder is melted in order to wet the parts of the joint, where it adheres to and connects the pieces after cooling. Metals or alloys suitable for use as solder should have a lower melting point than the pieces to be joined. The solder should also be resistant to oxidative and corrosive effects that would degrade the joint over time. Solder used in making electrical connections also needs to have favorable electrical characteristics. Soft solder typically has a melting point range of , and is commonly used in electronics, plumbing, and sheet metal work. Alloys that melt between are the most commonly used. Soldering performed using alloys with a melting point above is called "hard soldering", "silver soldering", or brazing. In specific proportions, some alloys are eutectic — that is, the alloy's melting point is the lowest possible for a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semidirect Product
In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. It is usually denoted with the symbol . There are two closely related concepts of semidirect product: * an ''inner'' semidirect product is a particular way in which a group can be made up of two subgroups, one of which is a normal subgroup. * an ''outer'' semidirect product is a way to construct a new group from two given groups by using the Cartesian product as a set and a particular multiplication operation. As with direct products, there is a natural equivalence between inner and outer semidirect products, and both are commonly referred to simply as ''semidirect products''. For finite groups, the Schur–Zassenhaus theorem provides a sufficient condition for the existence of a decomposition as a semidirect product (also known as splitting extension). Inner semidirect product definitions Given a group with identity element , a subgroup , and a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
General Relativity
General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. General theory of relativity, relativity generalizes special relativity and refines Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time in physics, time, or four-dimensional spacetime. In particular, the ''curvature of spacetime'' is directly related to the energy and momentum of whatever is present, including matter and radiation. The relation is specified by the Einstein field equations, a system of second-order partial differential equations. Newton's law of universal gravitation, which describes gravity in classical mechanics, can be seen as a prediction of general relativity for the almost flat spacetime geometry around stationary mass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tetrad (general Relativity)
In general relativity, a frame field (also called a tetrad or vierbein) is a set of four pointwise- orthonormal vector fields, one timelike and three spacelike, defined on a Lorentzian manifold that is physically interpreted as a model of spacetime. The timelike unit vector field is often denoted by \vec_0 and the three spacelike unit vector fields by \vec_1, \vec_2, \, \vec_3. All tensorial quantities defined on the manifold can be expressed using the frame field and its dual coframe field. Frame fields were introduced into general relativity by Albert Einstein in 1928 and by Hermann Weyl in 1929.Hermann Weyl "Elektron und Gravitation I", ''Zeitschrift Physik'', 56, p330–352, 1929. The index notation for tetrads is explained in tetrad (index notation). Physical interpretation Frame fields of a Lorentzian manifold always correspond to a family of ideal observers immersed in the given spacetime; the integral curves of the timelike unit vector field are the worldlines of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vierbein
The tetrad formalism is an approach to general relativity that generalizes the choice of basis for the tangent bundle from a coordinate basis to the less restrictive choice of a local basis, i.e. a locally defined set of four linearly independent vector fields called a '' tetrad'' or ''vierbein''. It is a special case of the more general idea of a ''vielbein formalism'', which is set in (pseudo-)Riemannian geometry. This article as currently written makes frequent mention of general relativity; however, almost everything it says is equally applicable to (pseudo-)Riemannian manifolds in general, and even to spin manifolds. Most statements hold by substituting arbitrary n for n=4. In German, "" translates to "four", "" to "many", and "" to "leg". The general idea is to write the metric tensor as the product of two ''vielbeins'', one on the left, and one on the right. The effect of the vielbeins is to change the coordinate system used on the tangent manifold to one that is simp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sigma Model
In physics, a sigma model is a field theory that describes the field as a point particle confined to move on a fixed manifold. This manifold can be taken to be any Riemannian manifold, although it is most commonly taken to be either a Lie group or a symmetric space. The model may or may not be quantized. An example of the non-quantized version is the Skyrme model; it cannot be quantized due to non-linearities of power greater than 4. In general, sigma models admit (classical) topological soliton solutions, for example, the skyrmion for the Skyrme model. When the sigma field is coupled to a gauge field, the resulting model is described by Ginzburg–Landau theory. This article is primarily devoted to the classical field theory of the sigma model; the corresponding quantized theory is presented in the article titled " non-linear sigma model". Overview The name has roots in particle physics, where a sigma model describes the interactions of pions. Unfortunately, the "sigma meson" ... [...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 Koszu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Contorsion Tensor
The contorsion tensor in differential geometry is the difference between a connection with and without torsion in it. It commonly appears in the study of spin connections. Thus, for example, a vielbein together with a spin connection, when subject to the condition of vanishing torsion, gives a description of Einstein gravity. For supersymmetry, the same constraint, of vanishing torsion, gives (the field equations of) eleven-dimensional supergravity. That is, the contorsion tensor, along with the connection, becomes one of the dynamical objects of the theory, demoting the metric to a secondary, derived role. The elimination of torsion in a connection is referred to as the ''absorption of torsion'', and is one of the steps of Cartan's equivalence method for establishing the equivalence of geometric structures. Definition in metric geometry In metric geometry, the contorsion tensor expresses the difference between a metric-compatible affine connection with Christoffel symbol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Torsion Tensor
In differential geometry, the torsion tensor is a tensor that is associated to any affine connection. The torsion tensor is a bilinear map of two input vectors X,Y, that produces an output vector T(X,Y) representing the displacement within a tangent space when the tangent space is developed (or "rolled") along an infinitesimal parallelogram whose sides are X,Y. It is skew symmetric in its inputs, because developing over the parallelogram in the opposite sense produces the opposite displacement, similarly to how a screw moves in opposite ways when it is twisted in two directions. Torsion is particularly useful in the study of the geometry of geodesics. Given a system of parametrized geodesics, one can specify a class of affine connections having those geodesics, but differing by their torsions. There is a unique connection which ''absorbs the torsion'', generalizing the Levi-Civita connection to other, possibly non-metric situations (such as Finsler geometry). The difference ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tautological One-form
In mathematics, the tautological one-form is a special 1-form defined on the cotangent bundle T^Q of a manifold Q. In physics, it is used to create a correspondence between the velocity of a point in a mechanical system and its momentum, thus providing a bridge between Lagrangian mechanics and Hamiltonian mechanics (on the manifold Q). The exterior derivative of this form defines a symplectic form giving T^Q the structure of a symplectic manifold. The tautological one-form plays an important role in relating the formalism of Hamiltonian mechanics and Lagrangian mechanics. The tautological one-form is sometimes also called the Liouville one-form, the Poincaré one-form, the canonical one-form, or the symplectic potential. A similar object is the canonical vector field on the tangent bundle. Definition in coordinates To define the tautological one-form, select a coordinate chart U on T^*Q and a canonical coordinate system on U. Pick an arbitrary point m \in T^*Q. By defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamiltonian Mechanics
In physics, Hamiltonian mechanics is a reformulation of Lagrangian mechanics that emerged in 1833. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta''. Both theories provide interpretations of classical mechanics and describe the same physical phenomena. Hamiltonian mechanics has a close relationship with geometry (notably, symplectic geometry and Poisson structures) and serves as a Hamilton–Jacobi equation, link between classical and quantum mechanics. Overview Phase space coordinates (''p'', ''q'') and Hamiltonian ''H'' Let (M, \mathcal L) be a Lagrangian mechanics, mechanical system with configuration space (physics), configuration space M and smooth Lagrangian_mechanics#Lagrangian, Lagrangian \mathcal L. Select a standard coordinate system (\boldsymbol,\boldsymbol) on M. The quantities \textstyle p_i(\boldsymbol,\boldsymbol,t) ~\stackrel~ / are called ''m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cotangent Bundle
In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This may be generalized to categories with more structure than smooth manifolds, such as complex manifolds, or (in the form of cotangent sheaf) algebraic varieties or schemes. In the smooth case, any Riemannian metric or symplectic form gives an isomorphism between the cotangent bundle and the tangent bundle, but they are not in general isomorphic in other categories. Formal definition via diagonal morphism There are several equivalent ways to define the cotangent bundle. One way is through a diagonal mapping Δ and germs. Let ''M'' be a smooth manifold and let ''M''×''M'' be the Cartesian product of ''M'' with itself. The diagonal mapping Δ sends a point ''p'' in ''M'' to the point (''p'',''p'') of ''M''×''M'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
