Electromagnetic stress–energy tensorIn relativistic physics, the electromagnetic stress–energy tensor is the contribution to the stress–energy tensor due to the electromagnetic field. The stress–energy tensor describes the flow of energy and momentum in spacetime. The electromagnetic stress–energy tensor contains the negative of the classical Maxwell stress tensor that governs the electromagnetic interactions. In free space and flat space–time, the electromagnetic stress–energy tensor in SI units is where is the electromagnetic tensor and where is the Minkowski metric tensor of metric signature (− + + +).
Gaussian integralThe Gaussian integral, also known as the Euler–Poisson integral, is the integral of the Gaussian function over the entire real line. Named after the German mathematician Carl Friedrich Gauss, the integral is Abraham de Moivre originally discovered this type of integral in 1733, while Gauss published the precise integral in 1809. The integral has a wide range of applications. For example, with a slight change of variables it is used to compute the normalizing constant of the normal distribution.
Elliptic integralIn integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (1750). Their name originates from their originally arising in connection with the problem of finding the arc length of an ellipse. Modern mathematics defines an "elliptic integral" as any function f which can be expressed in the form where R is a rational function of its two arguments, P is a polynomial of degree 3 or 4 with no repeated roots, and c is a constant.
Stress (mechanics)In continuum mechanics, stress is a physical quantity that describes forces present during deformation. An object being pulled apart, such as a stretched elastic band, is subject to tensile stress and may undergo elongation. An object being pushed together, such as a crumpled sponge, is subject to compressive stress and may undergo shortening. The greater the force and the smaller the cross-sectional area of the body on which it acts, the greater the stress. Stress has units of force per area, such as newtons per square meter (N/m2) or pascal (Pa).
Stress–energy tensorThe stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields. This density and flux of energy and momentum are the sources of the gravitational field in the Einstein field equations of general relativity, just as mass density is the source of such a field in Newtonian gravity.
Cauchy stress tensorIn continuum mechanics, the Cauchy stress tensor , true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components that completely define the state of stress at a point inside a material in the deformed state, placement, or configuration. The tensor relates a unit-length direction vector e to the traction vector T(e) across an imaginary surface perpendicular to e: or, The SI units of both stress tensor and traction vector are N/m2, corresponding to the stress scalar.
Tensor productIn mathematics, the tensor product of two vector spaces V and W (over the same field) is a vector space to which is associated a bilinear map that maps a pair to an element of denoted An element of the form is called the tensor product of v and w. An element of is a tensor, and the tensor product of two vectors is sometimes called an elementary tensor or a decomposable tensor. The elementary tensors span in the sense that every element of is a sum of elementary tensors.
Mixed tensorIn tensor analysis, a mixed tensor is a tensor which is neither strictly covariant nor strictly contravariant; at least one of the indices of a mixed tensor will be a subscript (covariant) and at least one of the indices will be a superscript (contravariant). A mixed tensor of type or valence , also written "type (M, N)", with both M > 0 and N > 0, is a tensor which has M contravariant indices and N covariant indices. Such a tensor can be defined as a linear function which maps an (M + N)-tuple of M one-forms and N vectors to a scalar.
Tensor contractionIn multilinear algebra, a tensor contraction is an operation on a tensor that arises from the natural pairing of a finite-dimensional vector space and its dual. In components, it is expressed as a sum of products of scalar components of the tensor(s) caused by applying the summation convention to a pair of dummy indices that are bound to each other in an expression. The contraction of a single mixed tensor occurs when a pair of literal indices (one a subscript, the other a superscript) of the tensor are set equal to each other and summed over.
GeometryGeometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a geometer. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts.
Tensor (intrinsic definition)In mathematics, the modern component-free approach to the theory of a tensor views a tensor as an abstract object, expressing some definite type of multilinear concept. Their properties can be derived from their definitions, as linear maps or more generally; and the rules for manipulations of tensors arise as an extension of linear algebra to multilinear algebra. In differential geometry an intrinsic geometric statement may be described by a tensor field on a manifold, and then doesn't need to make reference to coordinates at all.
Weyl equationIn physics, particularly in quantum field theory, the Weyl equation is a relativistic wave equation for describing massless spin-1/2 particles called Weyl fermions. The equation is named after Hermann Weyl. The Weyl fermions are one of the three possible types of elementary fermions, the other two being the Dirac and the Majorana fermions. None of the elementary particles in the Standard Model are Weyl fermions. Previous to the confirmation of the neutrino oscillations, it was considered possible that the neutrino might be a Weyl fermion (it is now expected to be either a Dirac or a Majorana fermion).
