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Mathematical Tools for Physics
Mathematical Tools for Physics

The Raman Effect
The Raman Effect

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Angle and Circle Characterizations of Tangential Quadrilaterals

Lecture Notes 18.5: Lorentz Transformation of EM Fields, the EM
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Sample pages 2 PDF

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ExamView - Geometry Chapter 12 Test SAMPLE.tst

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Pants decompositions of random surfaces

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Symplectic structures -- a new approach to geometry.

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Math Handbook of Formulas, Processes and Tricks Trigonometry

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Math Handbook of Formulas, Processes and Tricks

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Metric geometry of locally compact groups

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General Relativity, Black Holes, and Cosmology

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Metric geometry of locally compact groups

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Tangent circles in the hyperbolic disk - Rose

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Projective Geometry on Manifolds - UMD MATH

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Venzo de Sabbata

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SSLC - MATHEMATICS CHAPTER 10 CIRCLES ENGLISH VERSION

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Power Domains and Iterated Function Systems

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Higher Order Frequency-Domain Computational Electromagnetics , Senior Member, IEEE

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kucukarslan et al.

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EPH-classifications in Geometry, Algebra, Analysis and Arithmetic

... this foliation. If F is a 2-dimensional geodesic folition on a compact 4-manifold with negative Euler characteristic, then M is a fibration by spheres S 2 over a compact surface. Moreover one can choose this fibration in such a way that its fibers are either everywhere tangent or everywhere transver ...
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Electromagnetic Field Theory

lengths of geodesics on riemann surfaces with boundary
lengths of geodesics on riemann surfaces with boundary

THE GEOMETRIES OF 3
THE GEOMETRIES OF 3

1 2 3 4 5 ... 30 >

Metric tensor

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 it assigns a positive value to every nonzero vector. 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.With a holonomic basis 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.
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