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Tensor calculus has many applications in physics, engineering and computer science including elasticity, continuum mechanics, electromagnetism (see mathematical descriptions of the electromagnetic field), general relativity (see mathematics of general relativity), quantum field theory, and machine learning.
Vector and tensor calculus in general curvilinear coordinates is used in tensor analysis on four-dimensional curvilinear manifolds in general relativity, [8] in the mechanics of curved shells, [6] in examining the invariance properties of Maxwell's equations which has been of interest in metamaterials [9] [10] and in many other fields.
A metric tensor is a (symmetric) (0, 2)-tensor; it is thus possible to contract an upper index of a tensor with one of the lower indices of the metric tensor in the product. This produces a new tensor with the same index structure as the previous tensor, but with lower index generally shown in the same position of the contracted upper index.
Tullio Levi-Civita, ForMemRS [1] (English: / ˈ t ʊ l i oʊ ˈ l ɛ v i ˈ tʃ ɪ v ɪ t ə /, Italian: [ˈtulljo ˈlɛːvi ˈtʃiːvita]; 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.
Einstein's theory popularised the tensor calculus of Ricci and Levi-Civita and introduced the notation for a Riemannian metric, and for the Christoffel symbols, both coming from G in Gravitation. Élie Cartan helped reformulate the foundations of the differential geometry of smooth manifolds in terms of exterior calculus and the theory of ...
The Einstein field equations, = describe how this curvature is produced by matter and radiation, where G ab is the Einstein tensor, = written in terms of the Ricci tensor R ab and Ricci scalar R = R ab g ab, T ab is the stress–energy tensor and κ = 8πG/c 4 is a constant.
In mathematics, a tensor is a certain kind of geometrical entity and array concept. It generalizes the concepts of scalar , vector and linear operator , in a way that is independent of any chosen frame of reference .
Furthermore, from a more abstract standpoint, a tensor is simply "there" and its components of either kind are only calculational artifacts whose values depend on the chosen coordinates. The explanation in geometric terms is that a general tensor will have contravariant indices as well as covariant indices, because it has parts that live in the ...
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