Tensor calculus

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, it was used by Albert Einstein to develop his theory of general relativity. Contrasted with the infinitesimal calculus, tensor calculus allows presentation of physics equations in a form that is independent of the choice of coordinates on the manifold.

Tensor calculus has many real-life applications in physics and engineering, including elasticity, continuum mechanics, electromagnetism (see mathematical descriptions of the electromagnetic field), general relativity (see mathematics of general relativity), quantum field theory.

Working with a main proponent of the exterior calculus Elie Cartan, the influential geometer Shiing-Shen Chern nicely summarizes the role of tensor calculus[1]

In our subject of differential geometry, where you talk about manifolds, one difficulty is that the geometry is described by coordinates, but the coordinates do not have meaning. They are allowed to undergo transformation. And in order to handle this kind of situation, an important tool is the so-called tensor analysis, or Ricci calculus, which was new to mathematicians. In mathematics you have a function, you write down the function, you calculate, or you add, or you multiply, or you can differentiate. You have something very concrete. In geometry the geometric situation is described by numbers, but you can change your numbers arbitrarily. So to handle this, you need the Ricci calculus.

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Further reading[edit]

  • Dimitrienko, Yuriy (2002). Tensor Analysis and Nonlinear Tensor Functions. Kluwer Academic Publishers (Springer). ISBN 1-4020-1015-X.
  • Sokolnikoff, Ivan S (1951). Tensor Analysis: Theory and Applications to Geometry and Mechanics of Continua. Wiley. ISBN 0471810525.
  • A.I. Borisenko & I.E. Tarapov (1979). Vector and Tensor Analysis with Applications (2nd ed.). Dover. ISBN 0486638332.CS1 maint: uses authors parameter (link)
  • Itskov, Mikhail (2015). Tensor Algebra and Tensor Analysis for Engineers: With Applications to Continuum Mechanics. Springer; 2nd edition. ISBN 9783319163420.
  • Tyldesley, J. R. (1973). An introduction to Tensor Analysis: For Engineers and Applied Scientists. Longman. ISBN 0-582-44355-5.
  • Kay, D. C. (1988). Tensor Calculus. Schaum’s Outlines. McGraw Hill. ISBN 0-07-033484-6.
  • Grinfeld, P. (2014). Introduction to Tensor Analysis and the Calculus of Moving Surfaces. Springer. ISBN 1-4614-7866-9.

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