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Pseudo-tensor
In physics and mathematics, a pseudotensor is usually a quantity that transforms like a tensor under an orientation-preserving coordinate transformation (e.g. a proper rotation) but additionally changes sign under an orientation-reversing coordinate transformation (e.g., an improper rotation), which is a transformation that can be expressed as a proper rotation followed by reflection. This is a generalization of a pseudovector. To evaluate a tensor or pseudotensor sign, it has to be contracted with some vectors, as many as its rank is, belonging to the space where the rotation is made while keeping the tensor coordinates unaffected (differently from what one does in the case of a base change). Under improper rotation a pseudotensor and a proper tensor of the same rank will have different sign which depends on the rank being even or odd. Sometimes inversion of the axes is used as an example of an improper rotation to see the behaviour of a pseudotensor, but it works only if vector ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Physics
Physics is the natural science that studies matter, its 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." Physics is one of the most fundamental scientific disciplines, with its main goal being to understand how the universe behaves. "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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the plane and space curves and 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 manifolds. A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying str ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Connection
In differential geometry, an affine connection is a geometric object on a smooth manifold which ''connects'' nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space. Connections are among the simplest methods of defining differentiation of the sections of vector bundles. The notion of an affine connection has its roots in 19th-century geometry and tensor calculus, but was not fully developed until the early 1920s, by Élie Cartan (as part of his general theory of connections) and Hermann Weyl (who used the notion as a part of his foundations for general relativity). The terminology is due to Cartan and has its origins in the identification of tangent spaces in Euclidean space by translation: the idea is that a choice of affine connection makes a manifold look infinitesimally like Euclidean space not just smoothly, but as an affine space. On any manifold of positive dimension ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Christoffel Symbols
In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distances to be measured on that surface. In differential geometry, an affine connection can be defined without reference to a metric, and many additional concepts follow: parallel transport, covariant derivatives, geodesics, etc. also do not require the concept of a metric. However, when a metric is available, these concepts can be directly tied to the "shape" of the manifold itself; that shape is determined by how the tangent space is attached to the cotangent space by the metric tensor. Abstractly, one would say that the manifold has an associated (orthonormal) frame bundle, with each " frame" being a possible choice of a coordinate frame. An invariant metric implies that the structure group of the frame bundle is the orthogonal group . ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integrand
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with differentiation, integration is a fundamental, essential operation of calculus,Integral calculus is a very well established mathematical discipline for which there are many sources. See and , for example. and serves as a tool to solve problems in mathematics and physics involving the area of an arbitrary shape, the length of a curve, and the volume of a solid, among others. The integrals enumerated here are those termed definite integrals, which can be interpreted as the signed area of the region in the plane that is bounded by the graph of a given function between two points in the real line. Conventionally, areas above the horizontal axis of the plane are positive while areas below are negative. Integrals also refer to the concept of an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobian Matrix And Determinant
In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables as input as the number of vector components of its output, its determinant is referred to as the Jacobian determinant. Both the matrix and (if applicable) the determinant are often referred to simply as the Jacobian in literature. Suppose is a function such that each of its first-order partial derivatives exist on . This function takes a point as input and produces the vector as output. Then the Jacobian matrix of is defined to be an matrix, denoted by , whose th entry is \mathbf J_ = \frac, or explicitly :\mathbf J = \begin \dfrac & \cdots & \dfrac \end = \begin \nabla^ f_1 \\ \vdots \\ \nabla^ f_m \end = \begin \dfrac & \cdots & \dfrac\\ \vdots & \ddots & \vdots\\ \dfrac & \cd ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible and the linear map represented by the matrix is an isomorphism. The determinant of a product of matrices is the product of their determinants (the preceding property is a corollary of this one). The determinant of a matrix is denoted , , or . The determinant of a matrix is :\begin a & b\\c & d \end=ad-bc, and the determinant of a matrix is : \begin a & b & c \\ d & e & f \\ g & h & i \end= aei + bfg + cdh - ceg - bdi - afh. The determinant of a matrix can be defined in several equivalent ways. Leibniz formula expresses the determinant as a sum of signed products of matrix entries such that each summand is the product of different entries, and the number of these summands is n!, the factorial o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integration By Substitution
In calculus, integration by substitution, also known as ''u''-substitution, reverse chain rule or change of variables, is a method for evaluating integrals and antiderivatives. It is the counterpart to the chain rule for differentiation, and can loosely be thought of as using the chain rule "backwards". Substitution for a single variable Introduction Before stating the result rigorously, consider a simple case using indefinite integrals. Compute \textstyle\int(2x^3+1)^7(x^2)\,dx. Set u=2x^3+1. This means \textstyle\frac=6x^2, or in differential form, du=6x^2\,dx. Now :\int(2x^3 +1)^7(x^2)\,dx = \frac\int\underbrace_\underbrace_=\frac\int u^\,du=\frac\left(\fracu^\right)+C=\frac(2x^3+1)^+C, where C is an arbitrary constant of integration. This procedure is frequently used, but not all integrals are of a form that permits its use. In any event, the result should be verified by differentiating and comparing to the original integrand. :\frac\left frac(2x^3+1)^+C\r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Density On A Manifold
In mathematics, and specifically differential geometry, a density is a spatially varying quantity on a differentiable manifold that can be integrated in an intrinsic manner. Abstractly, a density is a section of a certain line bundle, called the density bundle. An element of the density bundle at ''x'' is a function that assigns a volume for the parallelotope spanned by the ''n'' given tangent vectors at ''x''. From the operational point of view, a density is a collection of functions on coordinate charts which become multiplied by the absolute value of the Jacobian determinant in the change of coordinates. Densities can be generalized into ''s''-densities, whose coordinate representations become multiplied by the ''s''-th power of the absolute value of the jacobian determinant. On an oriented manifold, 1-densities can be canonically identified with the ''n''-forms on ''M''. On non-orientable manifolds this identification cannot be made, since the density bundle is the tensor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Volume Element
In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates. Thus a volume element is an expression of the form :dV = \rho(u_1,u_2,u_3)\,du_1\,du_2\,du_3 where the u_i are the coordinates, so that the volume of any set B can be computed by :\operatorname(B) = \int_B \rho(u_1,u_2,u_3)\,du_1\,du_2\,du_3. For example, in spherical coordinates dV = u_1^2\sin u_2\,du_1\,du_2\,du_3, and so \rho = u_1^2\sin u_2. The notion of a volume element is not limited to three dimensions: in two dimensions it is often known as the area element, and in this setting it is useful for doing surface integrals. Under changes of coordinates, the volume element changes by the absolute value of the Jacobian determinant of the coordinate transformation (by the change of variables formula). This fact allows volume elements to be defined as a kind of measure on a manifold. On a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Volume Form
In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold M of dimension n, a volume form is an n-form. It is an element of the space of sections of the line bundle \textstyle^n(T^*M), denoted as \Omega^n(M). A manifold admits a nowhere-vanishing volume form if and only if it is orientable. An orientable manifold has infinitely many volume forms, since multiplying a volume form by a function yields another volume form. On non-orientable manifolds, one may instead define the weaker notion of a density. A volume form provides a means to define the integral of a function on a differentiable manifold. In other words, a volume form gives rise to a measure with respect to which functions can be integrated by the appropriate Lebesgue integral. The absolute value of a volume form is a volume element, which is also known variously as a ''twisted volume form'' or ''pseudo-volume form''. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
