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v. t. e. In mathematics, tensor calculus, tensor analysis, or Ricci calculus is an extension of vector calculus to tensor fields (tensors that may vary over a manifold, e.g. in spacetime). Developed by Gregorio Ricci-Curbastro and his student Tullio Levi-Civita, [1] it was used by Albert Einstein to develop his general theory of relativity.
In general relativity and tensor calculus, the contracted Bianchi identities are: [1] where is the Ricci tensor, the scalar curvature, and indicates covariant differentiation. These identities are named after Luigi Bianchi, although they had been already derived by Aurel Voss in 1880. [2] In the Einstein field equations, the contracted Bianchi ...
Schaum's Outlines (/ ʃɔːm /) is a series of supplementary texts for American high school, AP, and college-level courses, currently published by McGraw-Hill Education Professional, a subsidiary of McGraw-Hill Education. The outlines cover a wide variety of academic subjects including mathematics, engineering and the physical sciences ...
The symmetric quantity = 1 3 is called the fundamental (or metric) tensor of the Euclidean space in curvilinear coordinates. Note also that where hij are the Lamé coefficients. If we define the scale factors, hi, using we get a relation between the fundamental tensor and the Lamé coefficients.
Direction cosine. In analytic geometry, the direction cosines (or directional cosines) of a vector are the cosines of the angles between the vector and the three positive coordinate axes. Equivalently, they are the contributions of each component of the basis to a unit vector in that direction.
Biography. Ayres earned his Bachelor of Science degree from Washington College, Maryland and his master's and doctoral degrees from the University of Chicago. He taught during 1921–4 at Ogden College and another four years at Texas A&M before coming to Dickinson College in 1928. He was promoted to associate professor in June, 1935.
In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in Rn. More generally, tangent vectors are elements of a tangent space of a differentiable manifold. Tangent vectors can also be described in terms of ...
Ricci calculus. In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields on a differentiable manifold, with or without a metric tensor or connection. [a][1][2][3] It is also the modern name for what used to be called the absolute differential calculus (the foundation of tensor ...
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