What is Tensor: Definition and 1000 Discussions

In mathematics, a tensor is an algebraic object that describes a (multilinear) relationship between sets of algebraic objects related to a vector space. Objects that tensors may map between include vectors and scalars, and even other tensors. There are many types of tensors, including scalars and vectors (which are the simplest tensors), dual vectors, multilinear maps between vector spaces, and even some operations such as the dot product. Tensors are defined independent of any basis, although they are often referred to by their components in a basis related to a particular coordinate system.
Tensors have become important in physics because they provide a concise mathematical framework for formulating and solving physics problems in areas such as mechanics (stress, elasticity, fluid mechanics, moment of inertia, ...), electrodynamics (electromagnetic tensor, Maxwell tensor, permittivity, magnetic susceptibility, ...), or general relativity (stress–energy tensor, curvature tensor, ...) and others. In applications, it is common to study situations in which a different tensor can occur at each point of an object; for example the stress within an object may vary from one location to another. This leads to the concept of a tensor field. In some areas, tensor fields are so ubiquitous that they are often simply called "tensors".
Tullio Levi-Civita and Gregorio Ricci-Curbastro popularised tensors in 1900 - continuing the earlier work of Bernhard Riemann and Elwin Bruno Christoffel and others - as part of the absolute differential calculus. The concept enabled an alternative formulation of the intrinsic differential geometry of a manifold in the form of the Riemann curvature tensor.

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  1. S

    Electromagnetic strength tensor

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  2. E

    I Is the Energy Stress Tensor of Dust Always Zero Inside a Moving Cloud?

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  3. C

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  4. P

    What are the tensor and series questions in this homework?

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  5. Diego Berdeja

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  6. ibkev

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  7. A

    I Ricci tensor for Schwarzschild metric

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  8. S

    I Definition of stress-energy tensor

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  9. A

    I Derivation of E.M. Stress Energy Tensor

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  10. S

    I How Do Tensor and Vector Notations Differ in Physics?

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  11. e2m2a

    B Mass & Stress-Energy Tensor: Why Not Explicitly?

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  12. V

    I (2,0) tensor is not a tensor product of two vectors?

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  13. Jianphys17

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  14. S

    I What does the Einstein tensor actually tell you?

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  15. redtree

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  16. redtree

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  17. nomadreid

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  18. A

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  19. O

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  21. A

    I Exploring the Ricci Tensor: Einstein Field Equations

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  22. G

    I Why the tensor product (historical question)?

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  23. chi_rho

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  24. mertcan

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  25. J

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  26. S

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  29. Math Amateur

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  30. Math Amateur

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  31. mertcan

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  32. J

    A Stress-Energy Tensor: Basics & Questions

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  33. M

    Deriving perfect fluid energy tensor from point particles

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  34. M

    A Conservation of Electromagnetic Energy-Momentum Tensor

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  35. Logic Cloud

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  36. H

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  37. H

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  38. M

    A Tensor Calculus and Divergence

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  39. W

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  40. M

    Deriving perfect fluid energy tensor from point particles

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  41. A

    Maxwell's Equations from EM field tensor

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  42. D

    A Interpretation of the EM tensor as a rotation matrix

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  43. mertcan

    A Why Do Torsion Tensor Derivations Differ Between Sources?

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  44. mertcan

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  45. B

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  46. Math Amateur

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  48. T

    I Derivation of equations using tensor

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  49. J

    A Is the Dual Vector in Wald's Abstract Tensor Notation a Contraction?

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  50. T

    A Question about properites of tensor product

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