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Four Gradient
In differential geometry, the four-gradient (or 4-gradient) \boldsymbol is the four-vector analogue of the gradient \vec from vector calculus. In special relativity and in quantum mechanics, the four-gradient is used to define the properties and relations between the various physical four-vectors and tensors. Notation This article uses the metric signature. SR and GR are abbreviations for special relativity and general relativity respectively. c indicates the speed of light in vacuum. \eta_ = \operatorname ,-1,-1,-1/math> is the flat spacetime metric of SR. There are alternate ways of writing four-vector expressions in physics: * The four-vector style can be used: \mathbf \cdot \mathbf, which is typically more compact and can use vector notation, (such as the inner product "dot"), always using bold uppercase to represent the four-vector, and bold lowercase to represent 3-space vectors, e.g. \vec \cdot \vec. Most of the 3-space vector rules have analogues in four-vector mathe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as classical antiquity, antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Nikolai Lobachevsky, Lobachevsky. The simplest examples of smooth spaces are the Differential geometry of curves, plane and space curves and Differential geometry of surfaces, surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries. Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Differentiation
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry. The partial derivative of a function f(x, y, \dots) with respect to the variable x is variously denoted by It can be thought of as the rate of change of the function in the x-direction. Sometimes, for the partial derivative of z with respect to x is denoted as \tfrac. Since a partial derivative generally has the same arguments as the original function, its functional dependence is sometimes explicitly signified by the notation, such as in: f'_x(x, y, \ldots), \frac (x, y, \ldots). The symbol used to denote partial derivatives is ∂. One of the first known uses of this symbol in mathematics is by Marquis de Condorcet from 1770, who used it for partial difference ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Four-current
In special and general relativity, the four-current (technically the four-current density) is the four-dimensional analogue of the current density, with units of charge per unit time per unit area. Also known as vector current, it is used in the geometric context of ''four-dimensional spacetime'', rather than separating time from three-dimensional space. Mathematically it is a four-vector and is Lorentz covariant. This article uses the summation convention for indices. See covariance and contravariance of vectors for background on raised and lowered indices, and raising and lowering indices on how to switch between them. Definition Using the Minkowski metric \eta_ of metric signature , the four-current components are given by: :J^\alpha = \left(c \rho, j^1 , j^2 , j^3 \right) = \left(c \rho, \mathbf \right) where: * is the speed of light; * is the volume charge density; * is the conventional current density; *The dummy index labels the spacetime dimensions. Motion of ch ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordinate is needed to specify a point on itfor example, the point at 5 on a number line. A surface, such as the boundary of a cylinder or sphere, has a dimension of two (2D) because two coordinates are needed to specify a point on itfor example, both a latitude and longitude are required to locate a point on the surface of a sphere. A two-dimensional Euclidean space is a two-dimensional space on the plane. The inside of a cube, a cylinder or a sphere is three-dimensional (3D) because three coordinates are needed to locate a point within these spaces. In classical mechanics, space and time are different categories and refer to absolute space and time. That conception of the world is a four-dimensional space but not the one that w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
4-position
In special relativity, a four-vector (or 4-vector, sometimes Lorentz vector) is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vector space considered as a representation space of the standard representation of the Lorentz group, the (,) representation. It differs from a Euclidean vector in how its magnitude is determined. The transformations that preserve this magnitude are the Lorentz transformations, which include spatial rotations and boosts (a change by a constant velocity to another inertial reference frame). Four-vectors describe, for instance, position in spacetime modeled as Minkowski space, a particle's four-momentum , the amplitude of the electromagnetic four-potential at a point in spacetime, and the elements of the subspace spanned by the gamma matrices inside the Dirac algebra. The Lorentz group may be represented by 4×4 matrices . The action ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Current Sources And Sinks
In the physical sciences, engineering and mathematics, sources and sinks is an analogy used to describe properties of vector fields. It generalizes the idea of fluid sources and sinks (like the Tap (valve), faucet and Drain (plumbing), drain of a bathtub) across different scientific disciplines. These terms describe points, regions, or entities where a vector field originates or terminates. This analogy is usually invoked when discussing the continuity equation, the divergence of the field and the divergence theorem. The analogy sometimes includes swirls and saddles for points that are neither of the two. In the case of electric fields the idea of flow is replaced by field lines and the sources and sinks are electric charges. Description and fluid dynamics analogy In physics, a vector field \mathbf b(x,y,z) is a function that returns a Vector (mathematics and physics), vector and is defined for each point (with coordinates x,y,z) in a region of space. The idea of sources and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
