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In mathematics, tensor calculus or tensor analysis 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 stress analysis, continuum mechanics, electromagnetism (see mathematical descriptions of the electromagnetic field), and general relativity (see mathematics of general relativity).

Vector analysis
Matrix calculus
Ricci calculus
Tensors in curvilinear coordinates
Multilinear subspace learning

Books

Dimitrienko, Yuriy (2002). Tensor Analysis and Nonlinear Tensor Functions. Kluwer Academic Publishers (Springer). ISBN 1-4020-1015-X.
J.R. Tyldesley (1973). An introduction to Tensor Analysis: For Engineers and Applied Scientists. Longman. ISBN 0-582-44355-5.
D.C. Kay (1988). Tensor Calculus. Schaum’s Outlines, McGraw Hill (USA). ISBN 0-07-033484-6.