Tensor calculus

Part of a series of articles about
Calculus
Fundamental theorem Limits of functions Continuity Mean value theorem Rolle's theorem
Differential Definitions Derivative (generalizations) Differential infinitesimal of a function total Concepts Differentiation notation Second derivative Third derivative Change of variables Implicit differentiation Related rates Taylor's theorem Rules and identities Sum Product Chain Power Quotient General Leibniz Faà di Bruno's formula
Integral Lists of integrals Definitions Antiderivative Integral (improper) Riemann integral Lebesgue integration Contour integration Integration by Parts Discs Cylindrical shells Substitution (trigonometric) Partial fractions Order Reduction formulae
Series Geometric (arithmetico-geometric) Harmonic Alternating Power Binomial Taylor Convergence tests Summand limit (term test) Ratio Root Integral Direct comparison Limit comparison Alternating series Cauchy condensation Dirichlet Abel
Vector Gradient Divergence Curl Laplacian Directional derivative Identities Theorems Divergence Gradient Green's Stokes'
Multivariable Formalisms Matrix Tensor Exterior Geometric Definitions Partial derivative Multiple integral Line integral Surface integral Volume integral Jacobian Hessian matrix
Specialized Fractional Malliavin Stochastic Variations

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).

See also

• 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. 
• P.Grinfeld (2014). Introduction to Tensor Analysis and the Calculus of Moving Surfaces. Springer. ISBN 1-4614-7866-9.