Tensor
Tullio Levi-Civita and Gregorio Ricci-Curbastro popularised tensors in 1900 – continuing the earlier work of Bernhard Riemann, Elwin Bruno Christoffel, and others – as part of the absolute differential calculus.
The concept enabled an alternative formulation of the intrinsic differential geometry of a manifold in the form of the Riemann curvature tensor.
Thus while Tij and T ij can both be expressed as n-by-n matrices, and are numerically related via index juggling, the difference in their transformation laws indicates it would be improper to add them together.
For example, the fact that a vector is the same object in different coordinate systems can be captured by the following equations, using the formulas defined above: where
is the Kronecker delta, which functions similarly to the identity matrix, and has the effect of renaming indices (j into k in this example).
This shows several features of the component notation: the ability to re-arrange terms at will (commutativity), the need to use different indices when working with multiple objects in the same expression, the ability to rename indices, and the manner in which contravariant and covariant tensors combine so that all instances of the transformation matrix and its inverse cancel, so that expressions like
The transformation law for how the matrix of components of a linear operator changes with the basis is consistent with the transformation law for a contravariant vector, so that the action of a linear operator on a contravariant vector is represented in coordinates as the matrix product of their respective coordinate representations.
There is an action of the general linear group on the set of all ordered bases of an n-dimensional vector space.
[8] In this approach, a type (p, q) tensor T is defined as a multilinear map, where V∗ is the corresponding dual space of covectors, which is linear in each of its arguments.
By applying a multilinear map T of type (p, q) to a basis {ej} for V and a canonical cobasis {εi} for V∗, a (p + q)-dimensional array of components can be obtained.
But, because T is linear in all of its arguments, the components satisfy the tensor transformation law used in the multilinear array definition.
[11] Tensor products can be defined in great generality – for example, involving arbitrary modules over a ring.
In some applications, it is the tensor product of Hilbert spaces that is intended, whose properties are the most similar to the finite-dimensional case.
[13] In many applications, especially in differential geometry and physics, it is natural to consider a tensor with components that are functions of the point in a space.
The transformation law may then be expressed in terms of partial derivatives of the coordinate functions, defining a coordinate transformation,[1] The concepts of later tensor analysis arose from the work of Carl Friedrich Gauss in differential geometry, and the formulation was much influenced by the theory of algebraic forms and invariants developed during the middle of the nineteenth century.
[19] In Ricci's notation, he refers to "systems" with covariant and contravariant components, which are known as tensor fields in the modern sense.
[16] In the 20th century, the subject came to be known as tensor analysis, and achieved broader acceptance with the introduction of Albert Einstein's theory of general relativity, around 1915.
[16] From about the 1920s onwards, it was realised that tensors play a basic role in algebraic topology (for example in the Künneth theorem).
Contraction of an upper with a lower index of an (n, m)-tensor produces an (n − 1, m − 1)-tensor; this corresponds to moving diagonally up and to the left on the table.
The numbers of, respectively, vectors: n (contravariant indices) and dual vectors: m (covariant indices) in the input and output of a tensor determine the type (or valence) of the tensor, a pair of natural numbers (n, m), which determine the precise form of the transformation law.
The abstract index notation is a way to write tensors such that the indices are no longer thought of as numerical, but rather are indeterminates.
In detail, the tensor quantifying stress in a 3-dimensional solid object has components that can be conveniently represented as a 3 × 3 array.
Tensors of higher order do however capture ideas important in science and engineering, as has been shown successively in numerous areas as they develop.
To a good approximation (for sufficiently weak fields, assuming no permanent dipole moments are present), P is given by a Taylor series in E whose coefficients are the nonlinear susceptibilities: Here
The mass, in kg, of a region Ω is obtained by multiplying ρ by the volume of the region Ω, or equivalently integrating the constant ρ over the region: where the Cartesian coordinates x, y, z are measured in m. If the units of length are changed into cm, then the numerical values of the coordinate functions must be rescaled by a factor of 100: The numerical value of the density ρ must then also transform by 100−3 m3/cm3 to compensate, so that the numerical value of the mass in kg is still given by integral of
More generally, if the Cartesian coordinates x, y, z undergo a linear transformation, then the numerical value of the density ρ must change by a factor of the reciprocal of the absolute value of the determinant of the coordinate transformation, so that the integral remains invariant, by the change of variables formula for integration.
Such a quantity that scales by the reciprocal of the absolute value of the determinant of the coordinate transition map is called a scalar density.
But this is not quite the most general linear transformation law that such an object may have: tensor densities are non-rational, but are still semisimple representations.
It is possible to attach an additional discrete invariant to each frame that incorporates this path dependence, and which turns out (locally) to have values of ±1.
[41] A spinor is an object that transforms like a tensor under rotations in the frame, apart from a possible sign that is determined by the value of this discrete invariant.
