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Coordinate-free
A coordinate-free, or component-free, treatment of a scientific theory A scientific theory is an explanation of an aspect of the universe, natural world that can be or that has been reproducibility, repeatedly tested and has corroborating evidence in accordance with the scientific method, using accepted protocol (s ... or mathematical topic develops its concepts on any form of manifold without reference to any particular coordinate system. Benefits Coordinate-free treatments generally allow for simpler systems of equations and inherently constrain certain types of inconsistency, allowing greater mathematical elegance at the cost of some abstraction from the detailed formulae needed to evaluate these equations within a particular system of coordinates. In addition to elegance, coordinate-free treatments are crucial in certain applications for proving that a given definition is well formulated. For example, for a vector space V with basis v_1, ..., v_n, it may be tempting to co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Background Independence
Background independence is a condition in theoretical physics that requires the defining equations of a theory to be independent of the actual shape of the spacetime and the value of various fields within the spacetime. In particular this means that it must be possible not to refer to a specific coordinate system—the theory must be coordinate-free. In addition, the different spacetime configurations (or backgrounds) should be obtained as different solutions of the underlying equations. Description Background independence is a loosely defined property of a theory of physics. Roughly speaking, it limits the number of mathematical structures used to describe space and time that are put in place "by hand". Instead, these structures are the result of dynamical equations, such as Einstein field equations, so that one can determine from first principles what form they should take. Since the form of the metric determines the result of calculations, a theory with background independence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coordinate System
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine and standardize the position of the points or other geometric elements on a manifold such as Euclidean space. The coordinates are not interchangeable; they are commonly distinguished by their position in an ordered tuple, or by a label, such as in "the ''x''-coordinate". The coordinates are taken to be real numbers in elementary mathematics, but may be complex numbers or elements of a more abstract system such as a commutative ring. The use of a coordinate system allows problems in geometry to be translated into problems about numbers and ''vice versa''; this is the basis of analytic geometry. Common coordinate systems Number line The simplest example of a coordinate system is the identification of points on a line with real numbers using the '' number line''. In this system, an arbitrary point ''O'' (the ''origin'') is chosen on a given line. The coordinate o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Component-free Treatment Of Tensors
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. The same is true in general relativity, of tensor fields describing a physical property. The component-free approach is also used extensively in abstract algebra and homological algebra, where tensors arise naturally. Definition via tensor products of vector spaces Given a finite set of vector spaces over a common field , one may form their tensor product , an element of which is termed a tensor. A tensor on the vector space is then defined to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Foundations Of Geometry
Foundations of geometry is the study of geometries as axiomatic systems. There are several sets of axioms which give rise to Euclidean geometry or to non-Euclidean geometry, non-Euclidean geometries. These are fundamental to the study and of historical importance, but there are a great many modern geometries that are not Euclidean which can be studied from this viewpoint. The term axiomatic geometry can be applied to any geometry that is developed from an axiom system, but is often used to mean Euclidean geometry studied from this point of view. The completeness and independence of general axiomatic systems are important mathematical considerations, but there are also issues to do with the teaching of geometry which come into play. Axiomatic systems Based on ancient Greek methods, an ''axiomatic system'' is a formal description of a way to establish the ''mathematical truth'' that flows from a fixed set of assumptions. Although applicable to any area of mathematics, geometry is t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
General Covariance
In theoretical physics, general covariance, also known as diffeomorphism covariance or general invariance, consists of the Invariant (physics), invariance of the ''form'' of physical laws under arbitrary Derivative, differentiable coordinate transformations. The essential idea is that coordinates do not exist ''a priori'' in nature, but are only artifices used in describing nature, and hence should play no role in the formulation of fundamental physical laws. While this concept is exhibited by general relativity, which describes the dynamics of spacetime, one should not expect it to hold in less fundamental theories. For matter fields taken to exist independently of the background, it is almost never the case that their equations of motion will take the same form in curved space that they do in flat space. Overview A physical law expressed in a generally covariant fashion takes the same mathematical form in all coordinate systems, and is usually expressed in terms of tensor fie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scientific Theory
A scientific theory is an explanation of an aspect of the universe, natural world that can be or that has been reproducibility, repeatedly tested and has corroborating evidence in accordance with the scientific method, using accepted protocol (science), protocols of observation, measurement, and evaluation of results. Where possible, theories are tested under controlled conditions in an experiment. In circumstances not amenable to experimental testing, theories are evaluated through principles of abductive reasoning. Established scientific theories have withstood rigorous scrutiny and embody scientific knowledge. A scientific theory differs from a scientific fact: a fact is an observation and a theory organizes and explains multiple observations. Furthermore, a theory is expected to make predictions which could be confirmed or refuted with addition observations. Stephen Jay Gould wrote that "...facts and theories are different things, not rungs in a hierarchy of increasing certai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Calculus
Vector calculus or vector analysis is a branch of mathematics concerned with the differentiation and integration of vector fields, primarily in three-dimensional Euclidean space, \mathbb^3. The term ''vector calculus'' is sometimes used as a synonym for the broader subject of multivariable calculus, which spans vector calculus as well as partial differentiation and multiple integration. Vector calculus plays an important role in differential geometry and in the study of partial differential equations. It is used extensively in physics and engineering, especially in the description of electromagnetic fields, gravitational fields, and fluid flow. Vector calculus was developed from the theory of quaternions by J. Willard Gibbs and Oliver Heaviside near the end of the 19th century, and most of the notation and terminology was established by Gibbs and Edwin Bidwell Wilson in their 1901 book, '' Vector Analysis'', though earlier mathematicians such as Isaac Newton pioneered ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coordinate Conditions
In general relativity, the laws of physics can be expressed in a generally covariant form. In other words, the description of the world as given by the laws of physics does not depend on our choice of coordinate systems. However, it is often useful to fix upon a particular coordinate system, in order to solve actual problems or make actual predictions. A coordinate condition selects such coordinate system(s). Indeterminacy in general relativity The Einstein field equations do not determine the metric uniquely, even if one knows what the metric tensor equals everywhere at an initial time. This situation is analogous to the failure of the Maxwell equations to determine the potentials uniquely. In both cases, the ambiguity can be removed by gauge fixing. Thus, coordinate conditions are a type of gauge condition. No coordinate condition is generally covariant, but many coordinate conditions are Lorentz covariant or rotationally covariant. Naively, one might think that coordinat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Change Of Basis
In mathematics, an ordered basis of a vector space of finite dimension allows representing uniquely any element of the vector space by a coordinate vector, which is a sequence of scalars called coordinates. If two different bases are considered, the coordinate vector that represents a vector on one basis is, in general, different from the coordinate vector that represents on the other basis. A change of basis consists of converting every assertion expressed in terms of coordinates relative to one basis into an assertion expressed in terms of coordinates relative to the other basis. Such a conversion results from the ''change-of-basis formula'' which expresses the coordinates relative to one basis in terms of coordinates relative to the other basis. Using matrices, this formula can be written :\mathbf x_\mathrm = A \,\mathbf x_\mathrm, where "old" and "new" refer respectively to the initially defined basis and the other basis, \mathbf x_\mathrm and \mathbf x_\mathrm are the ... [...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] |
Computer Graphics
Computer graphics deals with generating images and art with the aid of computers. Computer graphics is a core technology in digital photography, film, video games, digital art, cell phone and computer displays, and many specialized applications. A great deal of specialized hardware and software has been developed, with the displays of most devices being driven by graphics hardware, computer graphics hardware. It is a vast and recently developed area of computer science. The phrase was coined in 1960 by computer graphics researchers Verne Hudson and William Fetter of Boeing. It is often abbreviated as CG, or typically in the context of film as Computer-generated imagery, computer generated imagery (CGI). The non-artistic aspects of computer graphics are the subject of Computer graphics (computer science), computer science research. Some topics in computer graphics include user interface design, Sprite (computer graphics), sprite graphics, raster graphics, Rendering (computer graph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
