[4] Jan Arnoldus Schouten developed the modern notation and formalism for this mathematical framework, and made contributions to the theory, during its applications to general relativity and differential geometry in the early twentieth century.
[5] The basis of modern tensor analysis was developed by Bernhard Riemann in a paper from 1861.
The description of tensor fields and operations on them in terms of their components is the focus of the Ricci calculus.
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), quantum field theory, and machine learning.
Working with a main proponent of the exterior calculus Élie Cartan, the influential geometer Shiing-Shen Chern summarizes the role of tensor calculus:[7]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.
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.
So to handle this, you need the Ricci calculus.Where a distinction is to be made between the space-like basis elements and a time-like element in the four-dimensional spacetime of classical physics, this is conventionally done through indices as follows:[8] Some sources use 4 instead of 0 as the index value corresponding to time; in this article, 0 is used.
In the expression Ai, i is interpreted as an index ranging over the values 1, 2, 3, while the x, y, z subscripts are only labels, not variables.
In the context of spacetime, the index value 0 conventionally corresponds to the label t. Indices themselves may be labelled using diacritic-like symbols, such as a hat (ˆ), bar (¯), tilde (˜), or prime (′) as in: to denote a possibly different basis for that index.
An example is in Lorentz transformations from one frame of reference to another, where one frame could be unprimed and the other primed, as in: This is not to be confused with van der Waerden notation for spinors, which uses hats and overdots on indices to reflect the chirality of a spinor.
This table summarizes how the manipulation of covariant and contravariant indices fit in with invariance under a passive transformation between bases, with the components of each basis set in terms of the other reflected in the first column.
For instance, if is in four dimensions (that is, each index runs from 0 to 3 or from 1 to 4), then because there are three free indices (α, β, δ), there are 43 = 64 equations.
For example, two antisymmetrizing indices imply: while three antisymmetrizing indices imply: as for a more specific example, if F represents the electromagnetic tensor, then the equation represents Gauss's law for magnetism and Faraday's law of induction.
Coordinates are typically denoted by xμ, but do not in general form the components of a vector.
In flat spacetime with linear coordinatization, a tuple of differences in coordinates, Δxμ, can be treated as a contravariant vector.
This may be repeated (without adding further commas): These components do not transform covariantly, unless the expression being differentiated is a scalar.
Less common alternatives to the semicolon include a forward slash ( / )[15] or in three-dimensional curved space a single vertical bar ( | ).
The Γαβγ for a Levi-Civita connection in a coordinate basis are called Christoffel symbols of the second kind.
It does not depend on either a metric tensor or a connection: it requires only the structure of a differentiable manifold.
The Lie derivative of a type (r, s) tensor field T along (the flow of) a contravariant vector field Xρ may be expressed using a coordinate basis as[20] This derivative is characterized by the product rule and the fact that the Lie derivative of a contravariant vector field along itself is zero: The Kronecker delta is like the identity matrix when multiplied and contracted: The components δαβ are the same in any basis and form an invariant tensor of type (1, 1), i.e. the identity of the tangent bundle over the identity mapping of the base manifold, and so its trace is an invariant.
This can be generalized to get the commutator for two covariant derivatives of an arbitrary tensor as follows: which are often referred to as the Ricci identities.
[24] The metric tensor gαβ is used for lowering indices and gives the length of any space-like curve where γ is any smooth strictly monotone parameterization of the path.
It also gives the duration of any time-like curve where γ is any smooth strictly monotone parameterization of the trajectory.