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Acceleration Vector
In mathematics and physics, acceleration is the rate of change of velocity of a curve with respect to a given linear connection. This operation provides us with a measure of the rate and direction of the "bend". Formal definition Let be given a differentiable manifold M, considered as spacetime (not only space), with a connection \Gamma. Let \gamma \colon\R \to M be a curve in M with tangent vector, i.e. (spacetime) velocity, (\tau), with parameter \tau. The (spacetime) acceleration vector of \gamma is defined by \nabla_ , where \nabla denotes the covariant derivative associated to \Gamma. It is a covariant derivative along \gamma, and it is often denoted by :\nabla_ =\frac. With respect to an arbitrary coordinate system (x^), and with (\Gamma^_) being the components of the connection (i.e., covariant derivative \nabla_:=\nabla_) relative to this coordinate system, defined by :\nabla_\frac= \Gamma^_\frac, for the acceleration vector field a^:=(\nabla_)^ one gets: :a^=v^\nabla ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Physics
Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events." It is one of the most fundamental scientific disciplines. "Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of physics. (...) You will come to see physics as a towering achievement of ... [...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] |
Differentiable Manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may then apply ideas from calculus while working within the individual charts, since each chart lies within a vector space to which the usual rules of calculus apply. If the charts are suitably compatible (namely, the transition from one chart to another is differentiable), then computations done in one chart are valid in any other differentiable chart. In formal terms, a differentiable manifold is a topological manifold with a globally defined differential structure. Any topological manifold can be given a differential structure locally by using the homeomorphisms in its atlas and the standard differential structure on a vector space. To induce a global differential structure on the local coordinate systems induced by the homeomorphism ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tangent Vector
In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. More generally, tangent vectors are elements of a '' tangent space'' of a differentiable manifold. Tangent vectors can also be described in terms of germs. Formally, a tangent vector at the point x is a linear derivation of the algebra defined by the set of germs at x. Motivation Before proceeding to a general definition of the tangent vector, we discuss its use in calculus and its tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ... properties. Calculus Let \mathbf(t) be a parametric smooth curve. The tangent vector is given by \mathbf'(t) provided it exists and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Covariant Derivative
In mathematics and physics, covariance is a measure of how much two variables change together, and may refer to: Statistics * Covariance matrix, a matrix of covariances between a number of variables * Covariance or cross-covariance between two random variables or data sets * Autocovariance, the covariance of a signal with a time-shifted version of itself * Covariance function, a function giving the covariance of a random field with itself at two locations Algebra and geometry * A covariant (invariant theory) is a bihomogeneous polynomial in and the coefficients of some homogeneous form in that is invariant under some group of linear transformations. * Covariance and contravariance of vectors, properties of how vector coordinates change under a change of basis ** Covariant transformation, a rule that describes how certain physical entities change under a change of coordinate system * Covariance and contravariance of functors, properties of functors * General covariance ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Index Notation
Abstract index notation (also referred to as slot-naming index notation) is a mathematical notation for tensors and spinors that uses indices to indicate their types, rather than their components in a particular basis. The indices are mere placeholders, not related to any basis and, in particular, are non-numerical. Thus it should not be confused with the Ricci calculus. The notation was introduced by Roger Penrose as a way to use the formal aspects of the Einstein summation convention to compensate for the difficulty in describing tensor contraction, contractions and covariant derivative, covariant differentiation in modern abstract tensor notation, while preserving the explicit Covariance and contravariance of vectors, covariance of the expressions involved. Let V be a vector space, and V^* its dual space. Consider, for example, an order-2 Covariance and contravariance of vectors, covariant tensor h \in V^*\otimes V^*. Then h can be identified with a bilinear form on V. In ot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Acceleration
In mechanics, acceleration is the Rate (mathematics), rate of change of the velocity of an object with respect to time. Acceleration is one of several components of kinematics, the study of motion. Accelerations are Euclidean vector, vector quantities (in that they have Magnitude (mathematics), magnitude and Direction (geometry), direction). The orientation of an object's acceleration is given by the orientation of the ''net'' force acting on that object. The magnitude of an object's acceleration, as described by Newton's second law, is the combined effect of two causes: * the net balance of all external forces acting onto that object — magnitude is Direct proportionality, directly proportional to this net resulting force; * that object's mass, depending on the materials out of which it is made — magnitude is Inverse proportionality, inversely proportional to the object's mass. The International System of Units, SI unit for acceleration is metre per second squared (, \ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Covariant Derivative
In mathematics and physics, covariance is a measure of how much two variables change together, and may refer to: Statistics * Covariance matrix, a matrix of covariances between a number of variables * Covariance or cross-covariance between two random variables or data sets * Autocovariance, the covariance of a signal with a time-shifted version of itself * Covariance function, a function giving the covariance of a random field with itself at two locations Algebra and geometry * A covariant (invariant theory) is a bihomogeneous polynomial in and the coefficients of some homogeneous form in that is invariant under some group of linear transformations. * Covariance and contravariance of vectors, properties of how vector coordinates change under a change of basis ** Covariant transformation, a rule that describes how certain physical entities change under a change of coordinate system * Covariance and contravariance of functors, properties of functors * General covariance ... [...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] |
