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, [1] Tensor calculus was used by Albert Einstein to develop his general theory of relativity. Unlike 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 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 and introduction to the mathematics of general relativity), quantum field theory, and machine learning.

General covariance:
In theoretical physics, general covariance, also known as diffeomorphism covariance or general invariance, consists of the invariance of the form of physical laws under arbitrary differentiable coordinate transformations. The essential idea is that coordinates do not exist a priori in nature, but are only artifices used in describing nature, and hence should play no role in the formulation of fundamental physical laws.

REMARK: Mathematics was not sufficiently refined in 1917 to cleave apart the demands for "no prior geometry" and for a geometric, coordinate-independent formulation of physics. Einstein described both demands by a single phrase, "general covariance". The "no prior geometry" demand actually fathered general relativity, but by doing so anonymously, disguised as "general covariance", it also fathered half a century of confusion.

A more modern interpretation of the physical content of the original principle of general covariance is that the Lie group $ {GL}_4(\mathbb{R}) $ is a fundamental "external" symmetry of the world. Other symmetries, including "internal" symmetries based on compact groups, now play a major role in fundamental physical theories.

Working with a main proponent of the differential forms of Élie Cartan,
the influential geometer Shiing-Shen Chern summarizes the role of tensor calculus [2] :

"In our subject of differential geometry, where you talk about a manifold, 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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• Lie group - Lie algebra correspondence
• Tensor
• Template:Tensors
• Glossary of tensor theory
• Template:Linear algebra
• Multilinear subspace learning
• Mathematics of general relativity.
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Content:
• 1 Syntax
• 2 Key concepts
• 2.1 Vector decomposition
• 2.1.1 Covariant vector decomposition
• 2.1.2 Contravariant vector decomposition
• 2.2 Metric tensor
• 2.3 Jacobian
• 2.4 Gradient vector
• 3 See also
• 4 References
• 5 Further reading
• 6 External links

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