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• Metric_tensor abstract "In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold (such as a surface in space) which takes as input a pair of tangent vectors v and w and produces a real number (scalar) g(v,w) in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space. In the same way as a dot product, metric tensors are used to define the length of, and angle between, tangent vectors.A metric tensor is called positive definite if every vector has positive length with respect to the metric. A manifold equipped with a positive definite metric tensor is known as a Riemannian manifold. By integration, the metric tensor allows one to define and compute the length of curves on the manifold. The curve connecting two points that (locally) has the smallest length is called a geodesic, and its length is the distance that a passenger in the manifold needs to traverse to go from one point to the other. Equipped with this notion of length, a Riemannian manifold is a metric space, meaning that it has a distance function d(p,q) whose value at a pair of points p and q is the distance from p to q. Conversely, the metric tensor itself is the derivative of the distance function (taken in a suitable manner). Thus the metric tensor gives the infinitesimal distance on the manifold.While the notion of a metric tensor was known in some sense to mathematicians such as Carl Gauss from the early 19th century, it was not until the early 20th century that its properties as a tensor were understood by, in particular, Gregorio Ricci-Curbastro and Tullio Levi-Civita who first codified the notion of a tensor. The metric tensor is an example of a tensor field, meaning that relative to a local coordinate system on the manifold, a metric tensor takes on the form of a symmetric matrix whose entries transform covariantly under changes to the coordinate system. Thus a metric tensor is a covariant symmetric tensor. From the coordinate-independent point of view, a metric tensor is defined to be a nondegenerate symmetric bilinear form on each tangent space that varies smoothly from point to point.".
• Metric_tensor wikiPageExternalLink index.php?id=11&PPN=PPN235181684_0054&DMDID=DMDLOG_0011&L=1.
• Metric_tensor wikiPageExternalLink digbib.cgi?PPN35283028X_0006_2NS.
• Metric_tensor wikiPageExternalLink numrel1.html.
• Metric_tensor wikiPageID "195795".
• Metric_tensor wikiPageRevisionID "605559229".
• Metric_tensor hasPhotoCollection Metric_tensor.
• Metric_tensor subject Category:Concepts_in_physics.
• Metric_tensor subject Category:Differential_geometry.
• Metric_tensor subject Category:Metric_tensors.
• Metric_tensor subject Category:Riemannian_geometry.
• Metric_tensor subject Category:Tensors.
• Metric_tensor type Abstraction100002137.
• Metric_tensor type Cognition100023271.
• Metric_tensor type Concept105835747.
• Metric_tensor type Content105809192.
• Metric_tensor type FundamentalPhysicsConcepts.
• Metric_tensor type Idea105833840.
• Metric_tensor type MetricTensors.
• Metric_tensor type PsychologicalFeature100023100.
• Metric_tensor type Quantity105855125.
• Metric_tensor type Tensor105864481.
• Metric_tensor type Tensors.
• Metric_tensor type Variable105857459.
• Metric_tensor comment "In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold (such as a surface in space) which takes as input a pair of tangent vectors v and w and produces a real number (scalar) g(v,w) in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space.".
• Metric_tensor label "Metric tensor".
• Metric_tensor label "Metrische tensor".
• Metric_tensor label "Metrischer Tensor".
• Metric_tensor label "Tenseur métrique".
• Metric_tensor label "Tensor metryczny".
• Metric_tensor label "Tensor métrico".
• Metric_tensor label "Tensor métrico".
• Metric_tensor label "Tensore metrico".
• Metric_tensor label "Метрический тензор".
• Metric_tensor label "度量张量".
• Metric_tensor label "計量テンソル".
• Metric_tensor sameAs Metrický_tenzor.
• Metric_tensor sameAs Metrischer_Tensor.
• Metric_tensor sameAs Tensor_métrico.
• Metric_tensor sameAs Tenseur_métrique.
• Metric_tensor sameAs Tensore_metrico.
• Metric_tensor sameAs 計量テンソル.
• Metric_tensor sameAs 계량_텐서.
• Metric_tensor sameAs Metrische_tensor.
• Metric_tensor sameAs Tensor_metryczny.
• Metric_tensor sameAs Tensor_métrico.
• Metric_tensor sameAs m.01btpw.
• Metric_tensor sameAs Q757269.
• Metric_tensor sameAs Q757269.
• Metric_tensor sameAs Metric_tensor.
• Metric_tensor wasDerivedFrom Metric_tensor?oldid=605559229.
• Metric_tensor isPrimaryTopicOf Metric_tensor.