Tensor

 ( Concepts In Physics ) 

In mathematics, a tensor is an algebraic object that describes a Multilinear map relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensors. There are many types of tensors, including scalars and vectors (which are the simplest tensors), , between vector spaces, and even some operations such as the dot product. Tensors are defined independent of any basis, although they are often referred to by their components in a basis related to a particular coordinate system; those components form an array, which can be thought of as a high-dimensional matrix.

Tensors have become important in physics because they provide a concise mathematical framework for formulating and solving physics problems in areas such as mechanics (stress, elasticity, quantum mechanics, fluid mechanics, moment of inertia, ...), electrodynamics (electromagnetic tensor, Maxwell tensor, permittivity, magnetic susceptibility, ...), and general relativity (stress–energy tensor, curvature tensor, ...). In applications, it is common to study situations in which a different tensor can occur at each point of an object; for example the stress within an object may vary from one location to another. This leads to the concept of a tensor field. In some areas, tensor fields are so ubiquitous that they are often simply called "tensors".

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.

The total number of indices () required to identify each component uniquely is equal to the dimension or the number of ways of an array, which is why a tensor is sometimes referred to as an -dimensional array or an -way array. The total number of indices is also called the order, degree or rank of a tensor,

Just as the components of a vector change when we change the basis of the vector space, the components of a tensor also change under such a transformation. Each type of tensor comes equipped with a transformation law that details how the components of the tensor respond to a change of basis. The components of a vector can respond in two distinct ways to a change of basis (see Covariance and contravariance of vectors), where the new basis vectors $\backslash mathbf\{\backslash hat\{e\}\}\_i$ are expressed in terms of the old basis vectors $\backslash mathbf\{e\}\_j$ as,

$\backslash mathbf\{\backslash hat\{e\}\}\_i\; =\; \backslash sum\_\{j=1\}^n\; \backslash mathbf\{e\}\_j\; R^j\_i\; =\; \backslash mathbf\{e\}\_j\; R^j\_i\; .$

Here R ^j _i are the entries of the change of basis matrix, and in the rightmost expression the summation sign was suppressed: this is the Einstein summation convention, which will be used throughout this article.The Einstein summation convention, in brief, requires the sum to be taken over all values of the index whenever the same symbol appears as a subscript and superscript in the same term. For example, under this convention $B\_i\; C^i\; =\; B\_1\; C^1\; +\; B\_2\; C^2\; +\; \backslash cdots\; +\; B\_n\; C^n$ The components v ⁱ of a column vector v transform with the matrix inverse of the matrix R,

$\backslash hat\{v\}^i\; =\; \backslash left(R^\{-1\}\backslash right)^i\_j\; v^j,$

where the hat denotes the components in the new basis. This is called a contravariant transformation law, because the vector components transform by the inverse of the change of basis. In contrast, the components, w _i, of a covector (or row vector), w, transform with the matrix R itself,

$\backslash hat\{w\}\_i\; =\; w\_j\; R^j\_i\; .$

This is called a covariant transformation law, because the covector components transform by the same matrix as the change of basis matrix. The components of a more general tensor are transformed by some combination of covariant and contravariant transformations, with one transformation law for each index. If the transformation matrix of an index is the inverse matrix of the basis transformation, then the index is called contravariant and is conventionally denoted with an upper index (superscript). If the transformation matrix of an index is the basis transformation itself, then the index is called covariant and is denoted with a lower index (subscript).

As a simple example, the matrix of a linear operator with respect to a basis is a rectangular array $T$ that transforms under a change of basis matrix $R\; =\; \backslash left(R^j\_i\backslash right)$ by $\backslash hat\{T\}\; =\; R^\{-1\}TR$. For the individual matrix entries, this transformation law has the form $\backslash hat\{T\}^\{i\text{'}\}\_\{j\text{'}\}\; =\; \backslash left(R^\{-1\}\backslash right)^\{i\text{'}\}\_i\; T^i\_j\; R^j\_\{j\text{'}\}$ so the tensor corresponding to the matrix of a linear operator has one covariant and one contravariant index: it is of type (1,1).

Combinations of covariant and contravariant components with the same index allow us to express geometric invariants. 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:

$\backslash mathbf\{v\}\; =\; \backslash hat\{v\}^i\; \backslash ,\backslash mathbf\{\backslash hat\{e\}\}\_i\; =$

\left( \left(R^{-1}\right)^i_j {v}^j \right) \left( \mathbf_k R^k_i \right) =
\left( \left(R^{-1}\right)^i_j R^k_i \right) {v}^j \mathbf_k =
\delta_j^k {v}^j  \mathbf_k = {v}^k \,\mathbf_k = {v}^i \,\mathbf_i
     

where $\backslash delta^k\_j$ 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 $\{v\}^i\; \backslash ,\backslash mathbf\_i$ can immediately be seen to be geometrically identical in all coordinate systems.

Similarly, a linear operator, viewed as a geometric object, does not actually depend on a basis: it is just a linear map that accepts a vector as an argument and produces another vector. 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. That is, the components $(Tv)^i$ are given by $(Tv)^i\; =\; T^i\_j\; v^j$. These components transform contravariantly, since

$\backslash left(\backslash widehat\{Tv\}\backslash right)^\{i\text{'}\}\; =\; \backslash hat\{T\}^\{i\text{'}\}\_\{j\text{'}\}\; \backslash hat\{v\}^\{j\text{'}\}\; =\; \backslash left\; \backslash left\; =\; \backslash left(R^\{-1\}\backslash right)^\{i\text{'}\}\_i\; (Tv)^i\; .$

The transformation law for an order tensor with p contravariant indices and q covariant indices is thus given as,

 \hat{T}^{i'_1, \ldots, i'_p}_{j'_1, \ldots, j'_q} = \left(R^{-1}\right)^{i'_1}_{i_1} \cdots \left(R^{-1}\right)^{i'_p}_{i_p}
     

 T^{i_1, \ldots, i_p}_{j_1, \ldots, j_q}
     

 R^{j_1}_{j'_1}\cdots R^{j_q}_{j'_q}.
     

Here the primed indices denote components in the new coordinates, and the unprimed indices denote the components in the old coordinates. Such a tensor is said to be of order or type . The terms "order", "type", "rank", "valence", and "degree" are all sometimes used for the same concept. Here, the term "order" or "total order" will be used for the total dimension of the array (or its generalization in other definitions), in the preceding example, and the term "type" for the pair giving the number of contravariant and covariant indices. A tensor of type is also called a -tensor for short.

This discussion motivates the following formal definition:

\mathbf{f} to each basis of an n-dimensional vector space such that, if we apply the change of basis

$\backslash mathbf\{f\}\backslash mapsto\; \backslash mathbf\{f\}\backslash cdot\; R\; =\; \backslash left(\; \backslash mathbf\{e\}\_i\; R^i\_1,\; \backslash dots,\; \backslash mathbf\{e\}\_i\; R^i\_n\; \backslash right)$

T^{i'_1\dots i'_p}_{j'_1\dots j'_q}[\mathbf{f} \cdot R] = \left(R^{-1}\right)^{i'_1}_{i_1} \cdots \left(R^{-1}\right)^{i'_p}_{i_p}
     

 T^{i_1, \ldots, i_p}_{j_1, \ldots, j_q}[\mathbf{f}]
     

 R^{j_1}_{j'_1}\cdots R^{j_q}_{j'_q} .
     

The definition of a tensor as a multidimensional array satisfying a transformation law traces back to the work of Ricci.

An equivalent definition of a tensor uses the representations of the general linear group. There is an action of the general linear group on the set of all ordered basis of an n-dimensional vector space. If $\backslash mathbf\; f\; =\; (\backslash mathbf\; f\_1,\; \backslash dots,\; \backslash mathbf\; f\_n)$ is an ordered basis, and $R\; =\; \backslash left(R^i\_j\backslash right)$ is an invertible $n\backslash times\; n$ matrix, then the action is given by

$\backslash mathbf\; fR\; =\; \backslash left(\backslash mathbf\; f\_i\; R^i\_1,\; \backslash dots,\; \backslash mathbf\; f\_i\; R^i\_n\backslash right).$

Let F be the set of all ordered bases. Then F is a principal homogeneous space for GL( n). Let W be a vector space and let $\backslash rho$ be a representation of GL( n) on W (that is, a group homomorphism $\backslash rho:\; \backslash text\{GL\}(n)\; \backslash to\; \backslash text\{GL\}(W)$). Then a tensor of type $\backslash rho$ is an equivariant map $T:\; F\; \backslash to\; W$. Equivariance here means that

$T(FR)\; =\; \backslash rho\backslash left(R^\{-1\}\backslash right)T(F).$

When $\backslash rho$ is a tensor representation of the general linear group, this gives the usual definition of tensors as multidimensional arrays. This definition is often used to describe tensors on manifolds, and readily generalizes to other groups.

$T:\; \backslash underbrace\{V^*\; \backslash times\backslash dots\backslash times\; V^*\}\_\{p\; \backslash text\{\; copies\}\}\; \backslash times\; \backslash underbrace\{\; V\; \backslash times\backslash dots\backslash times\; V\}\_\{q\; \backslash text\{\; copies\}\}\; \backslash rightarrow\; \backslash mathbb\{R\},$

where V^∗ is the corresponding dual space of covectors, which is linear in each of its arguments. The above assumes V is a vector space over the , . More generally, V can be taken over any field F (e.g. the ), with F replacing as the codomain of the multilinear maps.

By applying a multilinear map T of type to a basis { e _j} for V and a canonical cobasis { ε ⁱ} for V^∗,

$T^\{i\_1\backslash dots\; i\_p\}\_\{j\_1\backslash dots\; j\_q\}\; \backslash equiv\; T\backslash left(\backslash boldsymbol\{\backslash varepsilon\}^\{i\_1\},\; \backslash ldots,\backslash boldsymbol\{\backslash varepsilon\}^\{i\_p\},\; \backslash mathbf\{e\}\_\{j\_1\},\; \backslash ldots,\; \backslash mathbf\{e\}\_\{j\_q\}\backslash right),$

a -dimensional array of components can be obtained. A different choice of basis will yield different components. But, because T is linear in all of its arguments, the components satisfy the tensor transformation law used in the multilinear array definition. The multidimensional array of components of T thus form a tensor according to that definition. Moreover, such an array can be realized as the components of some multilinear map T. This motivates viewing multilinear maps as the intrinsic objects underlying tensors.

In viewing a tensor as a multilinear map, it is conventional to identify the double dual V^∗∗ of the vector space V, i.e., the space of linear functionals on the dual vector space V^∗, with the vector space V. There is always a natural linear map from V to its double dual, given by evaluating a linear form in V^∗ against a vector in V. This linear mapping is an isomorphism in finite dimensions, and it is often then expedient to identify V with its double dual.

A type tensor is defined in this context as an element of the tensor product of vector spaces,

$T\; \backslash in\; \backslash underbrace\{V\; \backslash otimes\backslash dots\backslash otimes\; V\}\_\{p\backslash text\{\; copies\}\}\; \backslash otimes\; \backslash underbrace\{V^*\; \backslash otimes\backslash dots\backslash otimes\; V^*\}\_\{q\; \backslash text\{\; copies\}\}.$

A basis of and basis of naturally induce a basis of the tensor product . The components of a tensor are the coefficients of the tensor with respect to the basis obtained from a basis for and its dual basis , i.e.

$T\; =\; T^\{i\_1\backslash dots\; i\_p\}\_\{j\_1\backslash dots\; j\_q\}\backslash ;\; \backslash mathbf\{e\}\_\{i\_1\}\backslash otimes\backslash cdots\backslash otimes\; \backslash mathbf\{e\}\_\{i\_p\}\backslash otimes\; \backslash boldsymbol\{\backslash varepsilon\}^\{j\_1\}\backslash otimes\backslash cdots\backslash otimes\; \backslash boldsymbol\{\backslash varepsilon\}^\{j\_q\}.$

Using the properties of the tensor product, it can be shown that these components satisfy the transformation law for a type tensor. Moreover, the universal property of the tensor product gives a bijection between tensors defined in this way and tensors defined as multilinear maps.

This 1 to 1 correspondence can be achieved in the following way, because in the finite-dimensional case there exists a canonical isomorphism between a vector space and its double dual:

$U\; \backslash otimes\; V\; \backslash cong\backslash left(U^\{*\; *\}\backslash right)\; \backslash otimes\backslash left(V^\{*\; *\}\backslash right)\; \backslash cong\backslash left(U^\{*\}\; \backslash otimes\; V^\{*\}\backslash right)^\{*\}\; \backslash cong\; \backslash operatorname\{Hom\}^\{2\}\backslash left(U^\{*\}\; \backslash times\; V^\{*\}\; ;\; \backslash mathbb\{F\}\backslash right)$

The last line is using the universal property of the tensor product, that there is a 1 to 1 correspondence between maps from $\backslash operatorname\{Hom\}^\{2\}\backslash left(U^\{*\}\; \backslash times\; V^\{*\}\; ;\; \backslash mathbb\{F\}\backslash right)$ and $\backslash operatorname\{Hom\}\backslash left(U^\{*\}\; \backslash otimes\; V^\{*\}\; ;\; \backslash mathbb\{F\}\backslash right)$.

Tensor products can be defined in great generality – for example, involving arbitrary modules over a ring. In principle, one could define a "tensor" simply to be an element of any tensor product. However, the mathematics literature usually reserves the term tensor for an element of a tensor product of any number of copies of a single vector space and its dual, as above.

 \hat{T}^{i'_1\dots i'_p}_{j'_1\dots j'_q}\left(\bar{x}^1, \ldots, \bar{x}^n\right) =
 \frac{\partial \bar{x}^{i'_1}}{\partial x^{i_1}}
 \cdots
 \frac{\partial \bar{x}^{i'_p}}{\partial x^{i_p}}
 \frac{\partial x^{j_1}}{\partial \bar{x}^{j'_1}}
 \cdots
 \frac{\partial x^{j_q}}{\partial \bar{x}^{j'_q}}
 T^{i_1\dots i_p}_{j_1\dots j_q}\left(x^1, \ldots, x^n\right).
     

Tensors and were also found to be useful in other fields such as continuum mechanics. Some well-known examples of tensors in differential geometry are such as , and the Riemann curvature tensor. The exterior algebra of Hermann Grassmann, from the middle of the nineteenth century, is itself a tensor theory, and highly geometric, but it was some time before it was seen, with the theory of differential forms, as naturally unified with tensor calculus. The work of Élie Cartan made differential forms one of the basic kinds of tensors used in mathematics, and Hassler Whitney popularized the tensor product.

From about the 1920s onwards, it was realised that tensors play a basic role in algebraic topology (for example in the Künneth theorem).

This table shows important examples of tensors on vector spaces and tensor fields on manifolds. The tensors are classified according to their type , where n is the number of contravariant indices, m is the number of covariant indices, and gives the total order of the tensor. For example, a bilinear form is the same thing as a -tensor; an inner product is an example of a -tensor, but not all -tensors are inner products. In the -entry of the table, M denotes the dimensionality of the underlying vector space or manifold because for each dimension of the space, a separate index is needed to select that dimension to get a maximally covariant antisymmetric tensor.

Raising an index on an -tensor produces an -tensor; this corresponds to moving diagonally down and to the left on the table. Symmetrically, lowering an index corresponds to moving diagonally up and to the right on the table. Contraction of an upper with a lower index of an -tensor produces an -tensor; this corresponds to moving diagonally up and to the left on the table.

Because the components of vectors and their duals transform differently under the change of their dual bases, there is a covariant and/or contravariant transformation law that relates the arrays, which represent the tensor with respect to one basis and that with respect to the other one. The numbers of, respectively, (contravariant indices) and dual (covariant indices) in the input and output of a tensor determine the type (or valence) of the tensor, a pair of natural numbers , which determine the precise form of the transformation law. The of a tensor is the sum of these two numbers.

The order (also degree or ) of a tensor is thus the sum of the orders of its arguments plus the order of the resulting tensor. This is also the dimensionality of the array of numbers needed to represent the tensor with respect to a specific basis, or equivalently, the number of indices needed to label each component in that array. For example, in a fixed basis, a standard linear map that maps a vector to a vector, is represented by a matrix (a 2-dimensional array), and therefore is a 2nd-order tensor. A simple vector can be represented as a 1-dimensional array, and is therefore a 1st-order tensor. Scalars are simple numbers and are thus 0th-order tensors. This way the tensor representing the scalar product, taking two vectors and resulting in a scalar has order , the same as the stress tensor, taking one vector and returning another . The mapping two vectors to one vector, would have order

The collection of tensors on a vector space and its dual forms a tensor algebra, which allows products of arbitrary tensors. Simple applications of tensors of order , which can be represented as a square matrix, can be solved by clever arrangement of transposed vectors and by applying the rules of matrix multiplication, but the tensor product should not be confused with this.

 (S \otimes T)^{i_1\ldots i_l i_{l+1}\ldots i_{l+n}}_{j_1\ldots j_k j_{k+1}\ldots j_{k+m}} =
 S^{i_1\ldots i_l}_{j_1\ldots j_k} T^{i_{l+1}\ldots i_{l+n}}_{j_{k+1}\ldots j_{k+m}}.
     

The contraction is often used in conjunction with the tensor product to contract an index from each tensor.

The contraction can also be understood using the definition of a tensor as an element of a tensor product of copies of the space V with the space V^∗ by first decomposing the tensor into a linear combination of simple tensors, and then applying a factor from V^∗ to a factor from V. For example, a tensor $T\; \backslash in\; V\backslash otimes\; V\backslash otimes\; V^*$ can be written as a linear combination

$T\; =\; v\_1\backslash otimes\; w\_1\backslash otimes\; \backslash alpha\_1\; +\; v\_2\backslash otimes\; w\_2\backslash otimes\; \backslash alpha\_2\; +\backslash cdots\; +\; v\_N\backslash otimes\; w\_N\backslash otimes\; \backslash alpha\_N.$

The contraction of T on the first and last slots is then the vector

$\backslash alpha\_1(v\_1)w\_1\; +\; \backslash alpha\_2(v\_2)w\_2\; +\; \backslash cdots\; +\; \backslash alpha\_N(v\_N)w\_N.$

In a vector space with an inner product (also known as a Metric tensor) g, the term contraction is used for removing two contravariant or two covariant indices by forming a trace with the metric tensor or its inverse. For example, a -tensor $T^\{ij\}$ can be contracted to a scalar through $T^\{ij\}\; g\_\{ij\}$ (yet again assuming the summation convention).

Conversely, the inverse operation can be defined, and is called raising an index. This is equivalent to a similar contraction on the product with a -tensor. This inverse metric tensor has components that are the matrix inverse of those of the metric tensor.

If a particular Volume form inside the material is singled out, the material on one side of the surface will apply a force on the other side. In general, this force will not be orthogonal to the surface, but it will depend on the orientation of the surface in a linear manner. This is described by a tensor of type , in linear elasticity, or more precisely by a tensor field of type , since the stresses may vary from point to point.

• Electromagnetic tensor (or Faraday tensor) in electromagnetism
• Finite deformation tensors for describing deformations and strain tensor for strain in continuum mechanics
• Permittivity and electric susceptibility are tensors in anisotropic media
• Four-tensors in general relativity (e.g. stress–energy tensor), used to represent momentum
• Spherical tensor operators are the eigenfunctions of the quantum angular momentum operator in spherical coordinates
• Diffusion tensors, the basis of diffusion tensor imaging, represent rates of diffusion in biological environments
• Quantum mechanics and quantum computing utilize tensor products for combination of quantum states

The field of nonlinear optics studies the changes to material polarization density under extreme electric fields. The polarization waves generated are related to the generating through the nonlinear susceptibility tensor. If the polarization P is not linearly proportional to the electric field E, the medium is termed nonlinear. 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:

$\backslash frac\{P\_i\}\{\backslash varepsilon\_0\}\; =\; \backslash sum\_j\; \backslash chi^\{(1)\}\_\{ij\}\; E\_j\; +\; \backslash sum\_\{jk\}\; \backslash chi\_\{ijk\}^\{(2)\}\; E\_j\; E\_k\; +\; \backslash sum\_\{jk\backslash ell\}\; \backslash chi\_\{ijk\backslash ell\}^\{(3)\}\; E\_j\; E\_k\; E\_\backslash ell\; +\; \backslash cdots.\; \backslash !$

Here $\backslash chi^\{(1)\}$ is the linear susceptibility, $\backslash chi^\{(2)\}$ gives the Pockels effect and second harmonic generation, and $\backslash chi^\{(3)\}$ gives the Kerr effect. This expansion shows the way higher-order tensors arise naturally in the subject matter.

$m\; =\; \backslash int\_\backslash Omega\; \backslash rho\backslash ,\; dx\backslash ,dy\backslash ,dz\; ,$

where the Cartesian coordinates , , are measured in . If the units of length are changed into , then the numerical values of the coordinate functions must be rescaled by a factor of 100:

$x\text{'}\; =\; 100\; x,\backslash quad\; y\text{'}\; =\; 100y,\backslash quad\; z\text{'}\; =\; 100\; z\; .$

The numerical value of the density must then also transform by to compensate, so that the numerical value of the mass in kg is still given by integral of $\backslash rho\backslash ,\; dx\backslash ,dy\backslash ,dz$. Thus $\backslash rho\text{'}\; =\; 100^\{-3\}\backslash rho$ (in units of ).

More generally, if the Cartesian coordinates , , undergo a linear transformation, then the numerical value of the density 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. To model a non-constant density, is a function of the variables , , (a scalar field), and under a curvilinear change of coordinates, it transforms by the reciprocal of the Jacobian of the coordinate change. For more on the intrinsic meaning, see Density on a manifold.

A tensor density transforms like a tensor under a coordinate change, except that it in addition picks up a factor of the absolute value of the determinant of the coordinate transition:

 T^{i'_1\dots i'_p}_{j'_1\dots j'_q}[\mathbf{f} \cdot R] =
   \left|\det R\right|^{-w}\left(R^{-1}\right)^{i'_1}_{i_1} \cdots \left(R^{-1}\right)^{i'_p}_{i_p}
 T^{i_1, \ldots, i_p}_{j_1, \ldots, j_q}[\mathbf{f}]
 R^{j_1}_{j'_1}\cdots R^{j_q}_{j'_q} .
     

Here is called the weight. In general, any tensor multiplied by a power of this function or its absolute value is called a tensor density, or a weighted tensor.

Under an affine transformation of the coordinates, a tensor transforms by the linear part of the transformation itself (or its inverse) on each index. These come from the rational representations of the general linear group. 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. A further class of transformations come from the logarithmic representation of the general linear group, a reducible but not semisimple representation, consisting of an with the transformation law

$(x,\; y)\; \backslash mapsto\; (x\; +\; y\backslash log\; \backslash left|\backslash det\; R\backslash right|,\; y).$

Spinors are elements of the spin representation of the rotation group, while tensors are elements of its tensor representations. Other have tensor representations, and so also tensors that are compatible with the group, but all non-compact classical groups have infinite-dimensional unitary representations as well.

• Array data type, for tensor storage and manipulation
• Bitensor

• Cartesian tensor
• Fibre bundle
• Glossary of tensor theory
• Multilinear projection
• One-form
• Tensor product of modules

• Application of tensor theory in engineering
• Continuum mechanics
• Covariant derivative
• Curvature
• Diffusion tensor MRI
• Einstein field equations
• Fluid mechanics
• Gravity
• Multilinear subspace learning
• Riemannian geometry
• Structure tensor
• Tensor Contraction Engine
• Tensor decomposition
• Tensor derivative
• Tensor software

• Richard L. Bishop (1980). 9780486640396, Dover. ISBN 9780486640396
• Donald A. Danielson (2025). 9780813340807, Westview (Perseus). ISBN 9780813340807
• Yuriy Dimitrienko (2025). 9781402010156, Springer. ISBN 9781402010156
• Nadir Jeevanjee (2025). 9780817647148, Birkhauser. . ISBN 9780817647148
• D. F. Lawden (2025). 9780486425405, Dover. ISBN 9780486425405
• (2025). 9789812383600, World Scientific. ISBN 9789812383600
• (1989). 9780486658407, Dover. ISBN 9780486658407
• James R. Munkres (1997). 9780813345482, Avalon. ISBN 9780813345482
  Chapter six gives a "from scratch" introduction to covariant tensors.
• David C Kay (1988). 9780070334847, McGraw-Hill. ISBN 9780070334847
• Bernard F. Schutz (1980). 9780521298872, Cambridge University Press. ISBN 9780521298872
• (1969). 9780486636122, Courier Corporation. ISBN 9780486636122

• (1976). 9780306375088, Plenum Press. ISBN 9780306375088
• (2025). 9780306375095 ISBN 9780306375095
• A discussion of the various approaches to teaching tensors, and recommendations of textbooks