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75 relations: Affine connection, Élie Cartan, Basis (linear algebra), Change of basis, Christoffel symbols, Comma, Connection (algebraic framework), Connection (mathematics), Connection (principal bundle), Connection (vector bundle), Connection form, Coordinate system, Cotangent space, Covariance and contravariance of vectors, Covariant transformation, Crelle's Journal, Curvature of Riemannian manifolds, Derivative, Differential geometry, Differential of a function, Differential operator, Directional derivative, Einstein notation, Elwin Bruno Christoffel, Euclidean space, Exterior covariant derivative, Foundations of Differential Geometry, Gauge covariant derivative, Gauge theory, General relativity, Geodesic, Gregorio Ricci-Curbastro, Hermann Weyl, Introduction to the mathematics of general relativity, Jacobian matrix and determinant, Jan Arnoldus Schouten, Jean-Louis Koszul, John Wiley & Sons, Levi-Civita connection, Lie algebra cohomology, Lie derivative, List of formulas in Riemannian geometry, Manifold, Mathematics, Metric tensor, Multilinear map, One-form, Orthonormality, Parallel transport, Partial derivative, ..., Physics, Principal part, Product rule, Projection (linear algebra), Pseudo-Riemannian manifold, Ricci calculus, Riemann curvature tensor, Riemannian geometry, Section (fiber bundle), Semicolon, Smoothness, Tangent bundle, Tangent vector, Tangential and normal components, Tensor, Tensor (intrinsic definition), Tensor density, Tensor derivative (continuum mechanics), Tensor field, Torsion tensor, Translation (geometry), Tullio Levi-Civita, Vector bundle, Vector calculus, Vector field. Expand index (25 more) »
Affine connection
In the branch of mathematics called 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.
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Élie Cartan
Élie Joseph Cartan, ForMemRS (9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups and their geometric applications.
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Basis (linear algebra)
In mathematics, a set of elements (vectors) in a vector space V is called a basis, or a set of, if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.
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Change of basis
In linear algebra, a basis for a vector space of dimension n is a set of n vectors, called basis vectors, with the property that every vector in the space can be expressed as a unique linear combination of the basis vectors.
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Christoffel symbols
In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection.
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Comma
The comma is a punctuation mark that appears in several variants in different languages.
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Connection (algebraic framework)
Geometry of quantum systems (e.g., noncommutative geometry and supergeometry) is mainly phrased in algebraic terms of modules and algebras.
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Connection (mathematics)
In geometry, the notion of a connection makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner.
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Connection (principal bundle)
In mathematics, a connection 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.
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Connection (vector bundle)
In mathematics, 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.
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Connection form
In mathematics, and specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving frames and differential forms.
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Coordinate system
In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space.
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Cotangent space
In differential geometry, one can attach to every point x of a smooth (or differentiable) manifold a vector space called the cotangent space at x. Typically, the cotangent space is defined as the dual space of the tangent space at x, although there are more direct definitions (see below).
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Covariance and contravariance of vectors
In multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis.
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Covariant transformation
In physics, a covariant transformation is a rule that specifies how certain entities, such as vectors or tensors, change under a change of basis.
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Crelle's Journal
Crelle's Journal, or just Crelle, is the common name for a mathematics journal, the Journal für die reine und angewandte Mathematik (in English: Journal for Pure and Applied Mathematics).
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Curvature of Riemannian manifolds
In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension greater than 2 is too complicated to be described by a single number at a given point.
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Derivative
The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value).
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Differential geometry
Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry.
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Differential of a function
In calculus, the differential represents the principal part of the change in a function y.
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Differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator.
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Directional derivative
In mathematics, the directional derivative of a multivariate differentiable function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity specified by v. It therefore generalizes the notion of a partial derivative, in which the rate of change is taken along one of the curvilinear coordinate curves, all other coordinates being constant.
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Einstein notation
In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention that implies summation over a set of indexed terms in a formula, thus achieving notational brevity.
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Elwin Bruno Christoffel
Elwin Bruno Christoffel (November 10, 1829 – March 15, 1900) was a German mathematician and physicist.
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Euclidean space
In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.
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Exterior covariant derivative
In mathematics, the exterior covariant derivative is an analog of an exterior derivative that takes into account the presence of a connection.
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Foundations of Differential Geometry
Foundations of Differential Geometry is an influential 2-volume mathematics book on differential geometry written by Shoshichi Kobayashi and Katsumi Nomizu.
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Gauge covariant derivative
The gauge covariant derivative is a variation of the covariant derivative used in general relativity.
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Gauge theory
In physics, a gauge theory is a type of field theory in which the Lagrangian is invariant under certain Lie groups of local transformations.
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General relativity
General relativity (GR, also known as the general theory of relativity or GTR) is the geometric theory of gravitation published by Albert Einstein in 1915 and the current description of gravitation in modern physics.
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Geodesic
In differential geometry, a geodesic is a generalization of the notion of a "straight line" to "curved spaces".
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Gregorio Ricci-Curbastro
Gregorio Ricci-Curbastro (12January 1925) was an Italian mathematician born in Lugo di Romagna.
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Hermann Weyl
Hermann Klaus Hugo Weyl, (9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher.
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Introduction to the mathematics of general relativity
The mathematics of general relativity is complex.
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Jacobian matrix and determinant
In vector calculus, the Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function.
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Jan Arnoldus Schouten
Jan Arnoldus Schouten (28 August 1883 – 20 January 1971) was a Dutch mathematician and Professor at the Delft University of Technology.
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Jean-Louis Koszul
Jean-Louis Koszul (January 3, 1921 – January 12, 2018) was a French mathematician, best known for studying geometry and discovering the Koszul complex.
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John Wiley & Sons
John Wiley & Sons, Inc., also referred to as Wiley, is a global publishing company that specializes in academic publishing.
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Levi-Civita connection
In Riemannian geometry, the Levi-Civita connection is a specific connection on the tangent bundle of a manifold.
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Lie algebra cohomology
In mathematics, Lie algebra cohomology is a cohomology theory for Lie algebras.
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Lie derivative
In differential geometry, the Lie derivative, named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar function, vector field and one-form), along the flow defined by another vector field.
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List of formulas in Riemannian geometry
This is a list of formulas encountered in Riemannian geometry.
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Manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.
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Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
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Metric tensor
In the mathematical field of differential geometry, a metric tensor is a type of function which takes as input a pair of tangent vectors and at a point of a surface (or higher dimensional differentiable manifold) and produces a real number scalar in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space.
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Multilinear map
In linear algebra, a multilinear map is a function of several variables that is linear separately in each variable.
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One-form
In linear algebra, a one-form on a vector space is the same as a linear functional on the space.
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Orthonormality
In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal and unit vectors.
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Parallel transport
In geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold.
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Partial derivative
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).
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Physics
Physics (from knowledge of nature, from φύσις phýsis "nature") is the natural science that studies matterAt the start of The Feynman Lectures on Physics, Richard Feynman offers the atomic hypothesis as the single most prolific scientific concept: "If, in some cataclysm, all scientific knowledge were to be destroyed one sentence what statement would contain the most information in the fewest words? I believe it is that all things are made up of atoms – little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another..." and its motion and behavior through space and time and that studies 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, and its main goal is 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. (...) You will come to see physics as a towering achievement of the human intellect in its quest to understand our world and ourselves."Physics is an experimental science. Physicists observe the phenomena of nature and try to find patterns that relate these phenomena.""Physics is the study of your world and the world and universe around you." Physics is one of the oldest academic disciplines and, through its inclusion of astronomy, perhaps the oldest. Over the last two millennia, physics, chemistry, biology, and certain branches of mathematics were a part of natural philosophy, but during the scientific revolution in the 17th century, these natural sciences emerged as unique research endeavors in their own right. Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, and the boundaries of physics are not rigidly defined. New ideas in physics often explain the fundamental mechanisms studied by other sciences and suggest new avenues of research in academic disciplines such as mathematics and philosophy. Advances in physics often enable advances in new technologies. For example, advances in the understanding of electromagnetism and nuclear physics led directly to the development of new products that have dramatically transformed modern-day society, such as television, computers, domestic appliances, and nuclear weapons; advances in thermodynamics led to the development of industrialization; and advances in mechanics inspired the development of calculus.
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Principal part
In mathematics, the principal part has several independent meanings, but usually refers to the negative-power portion of the Laurent series of a function.
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Product rule
In calculus, the product rule is a formula used to find the derivatives of products of two or more functions.
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Projection (linear algebra)
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that.
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Pseudo-Riemannian manifold
In differential geometry, a pseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a generalization of a Riemannian manifold in which the metric tensor need not be positive-definite, but need only be a non-degenerate bilinear form, which is a weaker condition.
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Ricci calculus
In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields.
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Riemann curvature tensor
In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common method used to express the curvature of Riemannian manifolds.
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Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point that varies smoothly from point to point.
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Section (fiber bundle)
In the mathematical field of topology, a section (or cross section) of a fiber bundle E is a continuous right inverse of the projection function \pi.
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Semicolon
The semicolon or semi colon is a punctuation mark that separates major sentence elements.
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Smoothness
In mathematical analysis, the smoothness of a function is a property measured by the number of derivatives it has that are continuous.
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Tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M. As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector.
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Tangent vector
In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point.
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Tangential and normal components
In mathematics, given a vector at a point on a curve, that vector can be decomposed uniquely as a sum of two vectors, one tangent to the curve, called the tangential component of the vector, and another one perpendicular to the curve, called the normal component of the vector.
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Tensor
In mathematics, tensors are geometric objects that describe linear relations between geometric vectors, scalars, and other tensors.
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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 multi-linear concept.
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Tensor density
In differential geometry, a tensor density or relative tensor is a generalization of the tensor field concept.
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Tensor derivative (continuum mechanics)
The derivatives of scalars, vectors, and second-order tensors with respect to second-order tensors are of considerable use in continuum mechanics.
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Tensor field
In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold).
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Torsion tensor
In differential geometry, the notion of torsion is a manner of characterizing a twist or screw of a moving frame around a curve.
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Translation (geometry)
In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure or a space by the same distance in a given direction.
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Tullio Levi-Civita
Tullio Levi-Civita, FRS (29 March 1873 – 29 December 1941) was an Italian mathematician, most famous for his work on absolute differential calculus (tensor calculus) and its applications to the theory of relativity, but who also made significant contributions in other areas.
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Vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every point x of the space X we associate (or "attach") a vector space V(x) in such a way that these vector spaces fit together to form another space of the same kind as X (e.g. a topological space, manifold, or algebraic variety), which is then called a vector bundle over X.
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Vector calculus
Vector calculus, or vector analysis, is a branch of mathematics concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3.
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Vector field
In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space.
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Absolute derivative, Absolute differentiation, Comma Derivative, Comma derivative, Contravariant derivative, Covariant differential, Covariant differentiation, Intrinsic derivative, Semicolon derivative, Tensor derivative.
References
[1] https://en.wikipedia.org/wiki/Covariant_derivative
