
In
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
and
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrix (mathemat ...
, a Cartesian tensor uses an
orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite Dimension (linear algebra), dimension is a Basis (linear algebra), basis for V whose vectors are orthonormal, that is, they are all unit vec ...
to
represent a
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...
in a
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
in the form of components. Converting a tensor's components from one such basis to another is done through an
orthogonal transformation
In linear algebra, an orthogonal transformation is a linear transformation ''T'' : ''V'' → ''V'' on a real inner product space ''V'', that preserves the inner product. That is, for each pair of elements of ''V'', we hav ...
.
The most familiar coordinate systems are the
two-dimensional
A two-dimensional space is a mathematical space with two dimensions, meaning points have two degrees of freedom: their locations can be locally described with two coordinates or they can move in two independent directions. Common two-dimension ...
and
three-dimensional
In geometry, a three-dimensional space (3D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values (''coordinates'') are required to determine the position (geometry), position of a point (geometry), poi ...
Cartesian coordinate
In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...
systems. Cartesian tensors may be used with any Euclidean space, or more technically, any finite-dimensional
vector space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
over the
field of
real number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s that has an
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, ofte ...
.
Use of Cartesian tensors occurs in
physics
Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
and
engineering
Engineering is the practice of using natural science, mathematics, and the engineering design process to Problem solving#Engineering, solve problems within technology, increase efficiency and productivity, and improve Systems engineering, s ...
, such as with the
Cauchy stress tensor
In continuum mechanics, the Cauchy stress tensor (symbol \boldsymbol\sigma, named after Augustin-Louis Cauchy), also called true stress tensor or simply stress tensor, completely defines the state of stress at a point inside a material in the d ...
and the
moment of inertia
The moment of inertia, otherwise known as the mass moment of inertia, angular/rotational mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is defined relatively to a rotational axis. It is the ratio between ...
tensor in
rigid body dynamics
In the physical science of dynamics, rigid-body dynamics studies the movement of systems of interconnected bodies under the action of external forces. The assumption that the bodies are '' rigid'' (i.e. they do not deform under the action ...
. Sometimes general
curvilinear coordinates
In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is invertible, l ...
are convenient, as in high-deformation
continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the deformation of and transmission of forces through materials modeled as a ''continuous medium'' (also called a ''continuum'') rather than as discrete particles.
Continuum mec ...
, or even necessary, as in
general relativity
General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
. While orthonormal bases may be found for some such coordinate systems (e.g.
tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...
to
spherical coordinates
In mathematics, a spherical coordinate system specifies a given point in three-dimensional space by using a distance and two angles as its three coordinates. These are
* the radial distance along the line connecting the point to a fixed point ...
), Cartesian tensors may provide considerable simplification for applications in which rotations of rectilinear coordinate axes suffice. The transformation is a
passive transformation, since the coordinates are changed and not the physical system.
Cartesian basis and related terminology
Vectors in three dimensions
In
3D Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
,
, the
standard basis
In mathematics, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as \mathbb^n or \mathbb^n) is the set of vectors, each of whose components are all zero, except one that equals 1. For exampl ...
is , , . Each basis vector points along the x-, y-, and z-axes, and the vectors are all
unit vector
In mathematics, a unit vector in a normed vector space is a Vector (mathematics and physics), vector (often a vector (geometry), spatial vector) of Norm (mathematics), length 1. A unit vector is often denoted by a lowercase letter with a circumfle ...
s (or normalized), so the basis is
orthonormal
In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal unit vectors. A unit vector means that the vector has a length of 1, which is also known as normalized. Orthogonal means that the vectors are all perpe ...
.
Throughout, when referring to
Cartesian coordinates
In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...
in
three dimensions, a right-handed system is assumed and this is much more common than a left-handed system in practice, see
orientation (vector space) for details.
For Cartesian tensors of order 1, a Cartesian vector can be written algebraically as a
linear combination
In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...
of the basis vectors , , :
where the
coordinate
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine and standardize the position of the points or other geometric elements on a manifold such as Euclidean space. The coordinates are ...
s of the vector with respect to the Cartesian basis are denoted , , . It is common and helpful to display the basis vectors as
column vector
In linear algebra, a column vector with elements is an m \times 1 matrix consisting of a single column of entries, for example,
\boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end.
Similarly, a row vector is a 1 \times n matrix for some , c ...
s
when we have a
coordinate vector
In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers (a tuple) that describes the vector in terms of a particular ordered basis. An easy example may be a position such as (5, 2, 1) in a 3-dimension ...
in a column vector representation:
A
row vector
In linear algebra, a column vector with elements is an m \times 1 matrix consisting of a single column of entries, for example,
\boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end.
Similarly, a row vector is a 1 \times n matrix for some , co ...
representation is also legitimate, although in the context of general curvilinear coordinate systems the row and column vector representations are used separately for specific reasons – see
Einstein notation
In mathematics, especially the usage of linear algebra in mathematical physics and differential geometry, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies ...
and
covariance and contravariance of vectors
In physics, especially in multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis. Briefly, a contravariant vecto ...
for why.
The term "component" of a vector is ambiguous: it could refer to:
*a specific coordinate of the vector such as (a scalar), and similarly for and , or
*the coordinate scalar-multiplying the corresponding basis vector, in which case the "-component" of is (a vector), and similarly for and .
A more general notation is
tensor index notation
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...
, which has the flexibility of numerical values rather than fixed coordinate labels. The Cartesian labels are replaced by tensor indices in the basis vectors , , and coordinates , , . In general, the notation , , refers to ''any'' basis, and , , refers to the corresponding coordinate system; although here they are restricted to the Cartesian system. Then:
It is standard to use the
Einstein notation
In mathematics, especially the usage of linear algebra in mathematical physics and differential geometry, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies ...
—the summation sign for summation over an index that is present exactly twice within a term may be suppressed for notational conciseness:
An advantage of the index notation over coordinate-specific notations is the independence of the dimension of the underlying vector space, i.e. the same expression on the right hand side takes the same form in higher dimensions (see below). Previously, the Cartesian labels x, y, z were just labels and ''not'' indices. (It is informal to say "''i'' = x, y, z").
Second-order tensors in three dimensions
A
dyadic tensor T is an order-2 tensor formed by the
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces V and W (over the same field) is a vector space to which is associated a bilinear map V\times W \rightarrow V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of ...
of two Cartesian vectors and , written . Analogous to vectors, it can be written as a linear combination of the tensor basis , , ..., (the right-hand side of each identity is only an abbreviation, nothing more):
Representing each basis tensor as a matrix:
then can be represented more systematically as a matrix:
See
matrix multiplication
In mathematics, specifically in linear algebra, matrix multiplication is a binary operation that produces a matrix (mathematics), matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the n ...
for the notational correspondence between matrices and the dot and tensor products.
More generally, whether or not is a tensor product of two vectors, it is always a linear combination of the basis tensors with coordinates , , ..., :
while in terms of tensor indices:
and in matrix form:
Second-order tensors occur naturally in physics and engineering when physical quantities have directional dependence in the system, often in a "stimulus-response" way. This can be mathematically seen through one aspect of tensors – they are
multilinear functions. A second-order tensor T which takes in a vector u of some magnitude and direction will return a vector v; of a different magnitude and in a different direction to u, in general. The notation used for
functions in
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...
leads us to write ,
[, used throughout] while the same idea can be expressed in matrix and index notations
[, see Appendix C.] (including the summation convention), respectively:
By "linear", if for two scalars and and vectors and , then in function and index notations:
and similarly for the matrix notation. The function, matrix, and index notations all mean the same thing. The matrix forms provide a clear display of the components, while the index form allows easier tensor-algebraic manipulation of the formulae in a compact manner. Both provide the physical interpretation of ''directions''; vectors have one direction, while second-order tensors connect two directions together. One can associate a tensor index or coordinate label with a basis vector direction.
The use of second-order tensors are the minimum to describe changes in magnitudes and directions of vectors, as the
dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a Scalar (mathematics), scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. N ...
of two vectors is always a scalar, while the
cross product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and ...
of two vectors is always a pseudovector perpendicular to the plane defined by the vectors, so these products of vectors alone cannot obtain a new vector of any magnitude in any direction. (See also below for more on the dot and cross products). The tensor product of two vectors is a second-order tensor, although this has no obvious directional interpretation by itself.
The previous idea can be continued: if takes in two vectors and , it will return a scalar . In function notation we write , while in matrix and index notations (including the summation convention) respectively:
The tensor T is linear in both input vectors. When vectors and tensors are written without reference to components, and indices are not used, sometimes a dot ⋅ is placed where summations over indices (known as
tensor contraction
In multilinear algebra, a tensor contraction is an operation on a tensor that arises from the canonical pairing of a vector space and its dual. In components, it is expressed as a sum of products of scalar components of the tensor(s) caused by ...
s) are taken. For the above cases:
[
motivated by the dot product notation:
More generally, a tensor of order which takes in vectors (where is between and inclusive) will return a tensor of order , see for further generalizations and details. The concepts above also apply to pseudovectors in the same way as for vectors. The vectors and tensors themselves can vary within throughout space, in which case we have ]vector field
In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space \mathbb^n. A vector field on a plane can be visualized as a collection of arrows with given magnitudes and dire ...
s and tensor field
In mathematics and physics, a tensor field is a function assigning a tensor to each point of a region of a mathematical space (typically a Euclidean space or manifold) or of the physical space. Tensor fields are used in differential geometry, ...
s, and can also depend on time.
Following are some examples:
For the electrical conduction example, the index and matrix notations would be:
while for the rotational kinetic energy :
See also constitutive equation
In physics and engineering, a constitutive equation or constitutive relation is a relation between two or more physical quantities (especially kinetic quantities as related to kinematic quantities) that is specific to a material or substance o ...
for more specialized examples.
Vectors and tensors in dimensions
In -dimensional Euclidean space over the real numbers, , the standard basis is denoted , , , ... . Each basis vector points along the positive axis, with the basis being orthonormal. Component of is given by the Kronecker delta
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:
\delta_ = \begin
0 &\text i \neq j, \\
1 &\ ...
:
A vector in takes the form:
Similarly for the order-2 tensor above, for each vector a and b in :
or more generally:
Transformations of Cartesian vectors (any number of dimensions)
Meaning of "invariance" under coordinate transformations
The position vector
In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents a point ''P'' in space. Its length represents the distance in relation to an arbitrary reference origin ''O'', and ...
in is a simple and common example of a vector, and can be represented in ''any'' coordinate system
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine and standardize the position of the points or other geometric elements on a manifold such as Euclidean space. The coordinates are ...
. Consider the case of rectangular coordinate systems with orthonormal bases only. It is possible to have a coordinate system with rectangular geometry if the basis vectors are all mutually perpendicular and not normalized, in which case the basis is ortho''gonal'' but not ortho''normal''. However, orthonormal bases are easier to manipulate and are often used in practice. The following results are true for orthonormal bases, not orthogonal ones.
In one rectangular coordinate system, as a contravector has coordinates and basis vectors , while as a covector it has coordinates and basis covectors , and we have:
In another rectangular coordinate system, as a contravector has coordinates and basis , while as a covector it has coordinates and basis , and we have:
Each new coordinate is a function of all the old ones, and vice versa for the inverse function
In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ .
For a function f\colon ...
:
and similarly each new basis vector is a function of all the old ones, and vice versa for the inverse function:
for all , .
A vector is invariant under any change of basis, so if coordinates transform according to a transformation matrix
In linear algebra, linear transformations can be represented by matrices. If T is a linear transformation mapping \mathbb^n to \mathbb^m and \mathbf x is a column vector with n entries, then there exists an m \times n matrix A, called the transfo ...
, the bases transform according to the matrix inverse
In linear algebra, an invertible matrix (''non-singular'', ''non-degenarate'' or ''regular'') is a square matrix that has an inverse. In other words, if some other matrix is multiplied by the invertible matrix, the result can be multiplied by an ...
, and conversely if the coordinates transform according to inverse , the bases transform according to the matrix . The difference between each of these transformations is shown conventionally through the indices as superscripts for contravariance and subscripts for covariance, and the coordinates and bases are linearly transformed according to the following rules:
where represents the entries of the transformation matrix
In linear algebra, linear transformations can be represented by matrices. If T is a linear transformation mapping \mathbb^n to \mathbb^m and \mathbf x is a column vector with n entries, then there exists an m \times n matrix A, called the transfo ...
(row number is and column number is ) and denotes the entries of the inverse matrix of the matrix .
If is an orthogonal transformation
In linear algebra, an orthogonal transformation is a linear transformation ''T'' : ''V'' → ''V'' on a real inner product space ''V'', that preserves the inner product. That is, for each pair of elements of ''V'', we hav ...
(orthogonal matrix
In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors.
One way to express this is
Q^\mathrm Q = Q Q^\mathrm = I,
where is the transpose of and is the identi ...
), the objects transforming by it are defined as Cartesian tensors. This geometrically has the interpretation that a rectangular coordinate system is mapped to another rectangular coordinate system, in which the norm
Norm, the Norm or NORM may refer to:
In academic disciplines
* Normativity, phenomenon of designating things as good or bad
* Norm (geology), an estimate of the idealised mineral content of a rock
* Norm (philosophy), a standard in normative e ...
of the vector is preserved (and distances are preserved).
The determinant
In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...
of is , which corresponds to two types of orthogonal transformation: () for rotations and () for improper rotation
In geometry, an improper rotation. (also called rotation-reflection, rotoreflection, rotary reflection,. or rotoinversion) is an isometry in Euclidean space that is a combination of a Rotation (geometry), rotation about an axis and a reflection ( ...
s (including reflection Reflection or reflexion may refer to:
Science and technology
* Reflection (physics), a common wave phenomenon
** Specular reflection, mirror-like reflection of waves from a surface
*** Mirror image, a reflection in a mirror or in water
** Diffuse r ...
s).
There are considerable algebraic simplifications, the matrix transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The tr ...
is the inverse
Inverse or invert may refer to:
Science and mathematics
* Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence
* Additive inverse, the inverse of a number that, when added to the ...
from the definition of an orthogonal transformation:
From the previous table, orthogonal transformations of covectors and contravectors are identical. There is no need to differ between raising and lowering indices
The asterisk ( ), from Late Latin , from Ancient Greek , , "little star", is a typographical symbol. It is so called because it resembles a conventional image of a heraldic star.
Computer scientists and mathematicians often vocalize it as st ...
, and in this context and applications to physics and engineering the indices are usually all subscripted to remove confusion for exponent
In mathematics, exponentiation, denoted , is an operation involving two numbers: the ''base'', , and the ''exponent'' or ''power'', . When is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, i ...
s. All indices will be lowered in the remainder of this article. One can determine the actual raised and lowered indices by considering which quantities are covectors or contravectors, and the relevant transformation rules.
Exactly the same transformation rules apply to any vector , not only the position vector. If its components do not transform according to the rules, is not a vector.
Despite the similarity between the expressions above, for the change of coordinates such as , and the action of a tensor on a vector like , is not a tensor, but is. In the change of coordinates, is a ''matrix'', used to relate two rectangular coordinate systems with orthonormal bases together. For the tensor relating a vector to a vector, the vectors and tensors throughout the equation all belong to the same coordinate system and basis.
Derivatives and Jacobian matrix elements
The entries of are partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). P ...
s of the new or old coordinates with respect to the old or new coordinates, respectively.
Differentiating with respect to :
so
is an element of the Jacobian matrix
In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. If this matrix is square, that is, if the number of variables equals the number of component ...
. There is a (partially mnemonical) correspondence between index positions attached to L and in the partial derivative: ''i'' at the top and ''j'' at the bottom, in each case, although for Cartesian tensors the indices can be lowered.
Conversely, differentiating with respect to :
so
is an element of the inverse Jacobian matrix, with a similar index correspondence.
Many sources state transformations in terms of the partial derivatives:
and the explicit matrix equations in 3d are:
similarly for
Projections along coordinate axes
As with all linear transformations, depends on the basis chosen. For two orthonormal bases
* projecting to the axes:
* projecting to the axes:
Hence the components reduce to direction cosine
In analytic geometry, the direction cosines (or directional cosines) of a vector are the cosines of the angles between the vector and the three positive coordinate axes. Equivalently, they are the contributions of each component of the basis t ...
s between the and axes:
where and are the angles between the and axes. In general, is not equal to , because for example and are two different angles.
The transformation of coordinates can be written:
and the explicit matrix equations in 3d are:
similarly for
The geometric interpretation is the components equal to the sum of projecting the components onto the axes.
The numbers arranged into a matrix would form a symmetric matrix
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally,
Because equal matrices have equal dimensions, only square matrices can be symmetric.
The entries of a symmetric matrix are symmetric with ...
(a matrix equal to its own transpose) due to the symmetry in the dot products, in fact it is the metric tensor
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
. By contrast or do ''not'' form symmetric matrices in general, as displayed above. Therefore, while the matrices are still orthogonal, they are not symmetric.
Apart from a rotation about any one axis, in which the and for some coincide, the angles are not the same as Euler angle
The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body with respect to a fixed coordinate system.Novi Commentarii academiae scientiarum Petropolitanae 20, 1776, pp. 189–207 (E478PDF/ref>
They ...
s, and so the matrices are not the same as the rotation matrices.
Transformation of the dot and cross products (three dimensions only)
The dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a Scalar (mathematics), scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. N ...
and cross product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and ...
occur very frequently, in applications of vector analysis to physics and engineering, examples include:
*power
Power may refer to:
Common meanings
* Power (physics), meaning "rate of doing work"
** Engine power, the power put out by an engine
** Electric power, a type of energy
* Power (social and political), the ability to influence people or events
Math ...
transferred by an object exerting a force with velocity along a straight-line path:
*tangential velocity
Velocity is a measurement of speed in a certain direction of motion. It is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of physical objects. Velocity is a vector (geometry), vector Physical q ...
at a point of a rotating rigid body
In physics, a rigid body, also known as a rigid object, is a solid body in which deformation is zero or negligible, when a deforming pressure or deforming force is applied on it. The distance between any two given points on a rigid body rema ...
with angular velocity
In physics, angular velocity (symbol or \vec, the lowercase Greek letter omega), also known as the angular frequency vector,(UP1) is a pseudovector representation of how the angular position or orientation of an object changes with time, i ...
:
*potential energy
In physics, potential energy is the energy of an object or system due to the body's position relative to other objects, or the configuration of its particles. The energy is equal to the work done against any restoring forces, such as gravity ...
of a magnetic dipole
In electromagnetism, a magnetic dipole is the limit of either a closed loop of electric current or a pair of poles as the size of the source is reduced to zero while keeping the magnetic moment constant.
It is a magnetic analogue of the Electri ...
of magnetic moment
In electromagnetism, the magnetic moment or magnetic dipole moment is the combination of strength and orientation of a magnet or other object or system that exerts a magnetic field. The magnetic dipole moment of an object determines the magnitude ...
in a uniform external magnetic field
A magnetic field (sometimes called B-field) is a physical field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular ...
:
*angular momentum
Angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of Momentum, linear momentum. It is an important physical quantity because it is a Conservation law, conserved quantity – the total ang ...
for a particle with position vector
In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents a point ''P'' in space. Its length represents the distance in relation to an arbitrary reference origin ''O'', and ...
and momentum
In Newtonian mechanics, momentum (: momenta or momentums; more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. ...
:
*torque
In physics and mechanics, torque is the rotational analogue of linear force. It is also referred to as the moment of force (also abbreviated to moment). The symbol for torque is typically \boldsymbol\tau, the lowercase Greek letter ''tau''. Wh ...
acting on an electric dipole
The electric dipole moment is a measure of the separation of positive and negative electrical charges within a system: that is, a measure of the system's overall polarity. The SI unit for electric dipole moment is the coulomb-metre (C⋅m). The ...
of electric dipole moment
The electric dipole moment is a measure of the separation of positive and negative electrical charges within a system: that is, a measure of the system's overall Chemical polarity, polarity. The International System of Units, SI unit for electric ...
in a uniform external electric field
An electric field (sometimes called E-field) is a field (physics), physical field that surrounds electrically charged particles such as electrons. In classical electromagnetism, the electric field of a single charge (or group of charges) descri ...
:
*induced surface current density
In electromagnetism, current density is the amount of charge per unit time that flows through a unit area of a chosen cross section. The current density vector is defined as a vector whose magnitude is the electric current per cross-sectional ...
in a magnetic material of magnetization
In classical electromagnetism, magnetization is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material. Accordingly, physicists and engineers usually define magnetization as the quanti ...
on a surface with unit normal
In geometry, a normal is an object (e.g. a line, ray, or vector) that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the infinite straight line perpendicular to the tangent line to the cur ...
:
How these products transform under orthogonal transformations is illustrated below.
Dot product, Kronecker delta, and metric tensor
The dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a Scalar (mathematics), scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. N ...
⋅ of each possible pairing of the basis vectors follows from the basis being orthonormal. For perpendicular pairs we have
while for parallel pairs we have
Replacing Cartesian labels by index notation as shown above
Above may refer to:
*Above (artist)
Tavar Zawacki (b. 1981, California) is a Polish, Portuguese - American abstract artist and
internationally recognized visual artist based in Berlin, Germany. From 1996 to 2016, he created work under the ...
, these results can be summarized by
where are the components of the Kronecker delta
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:
\delta_ = \begin
0 &\text i \neq j, \\
1 &\ ...
. The Cartesian basis can be used to represent in this way.
In addition, each metric tensor
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
component with respect to any basis is the dot product of a pairing of basis vectors:
For the Cartesian basis the components arranged into a matrix are:
so are the simplest possible for the metric tensor, namely the :
This is ''not'' true for general bases: orthogonal coordinates
In mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. ...
have diagonal
In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek � ...
metrics containing various scale factors (i.e. not necessarily 1), while general curvilinear coordinates
In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is invertible, l ...
could also have nonzero entries for off-diagonal components.
The dot product of two vectors and transforms according to
which is intuitive, since the dot product of two vectors is a single scalar independent of any coordinates. This also applies more generally to any coordinate systems, not just rectangular ones; the dot product in one coordinate system is the same in any other.
Cross product, Levi-Civita symbol, and pseudovectors
For the cross product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and ...
() of two vectors, the results are (almost) the other way round. Again, assuming a right-handed 3d Cartesian coordinate system, cyclic permutation
In mathematics, and in particular in group theory, a cyclic permutation is a permutation consisting of a single cycle. In some cases, cyclic permutations are referred to as cycles; if a cyclic permutation has ''k'' elements, it may be called a ''k ...
s in perpendicular directions yield the next vector in the cyclic collection of vectors:
while parallel vectors clearly vanish:
and replacing Cartesian labels by index notation as above
Above may refer to:
*Above (artist)
Tavar Zawacki (b. 1981, California) is a Polish, Portuguese - American abstract artist and
internationally recognized visual artist based in Berlin, Germany. From 1996 to 2016, he created work under the ...
, these can be summarized by:
where , , are indices which take values . It follows that:
These permutation relations and their corresponding values are important, and there is an object coinciding with this property: the Levi-Civita symbol
In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol or Levi-Civita epsilon represents a collection of numbers defined from the sign of a permutation of the natural numbers , for some ...
, denoted by . The Levi-Civita symbol entries can be represented by the Cartesian basis:
which geometrically corresponds to the volume
Volume is a measure of regions in three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch) ...
of a cube
A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...
spanned by the orthonormal basis vectors, with sign indicating orientation
Orientation may refer to:
Positioning in physical space
* Map orientation, the relationship between directions on a map and compass directions
* Orientation (housing), the position of a building with respect to the sun, a concept in building des ...
(and ''not'' a "positive or negative volume"). Here, the orientation is fixed by , for a right-handed system. A left-handed system would fix or equivalently .
The scalar triple product
In geometry and algebra, the triple product is a product of three 3- dimensional vectors, usually Euclidean vectors. The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vect ...
can now be written:
with the geometric interpretation of volume (of the parallelepiped
In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term ''rhomboid'' is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square.
Three equiva ...
spanned by , , ) and algebraically is a determinant
In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...
:
This in turn can be used to rewrite the cross product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and ...
of two vectors as follows:
Contrary to its appearance, the Levi-Civita symbol is ''not a tensor'', but a pseudotensor
In physics and mathematics, a pseudotensor is usually a quantity that transforms like a tensor under an orientation-preserving coordinate transformation (e.g. a proper rotation) but additionally changes sign under an orientation-reversing coordin ...
, the components transform according to:
Therefore, the transformation of the cross product of and is:
and so transforms as a pseudovector
In physics and mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under continuous rigid transformations such as rotations or translations, but which does ''not'' transform like a vector under certain ' ...
, because of the determinant factor.
The tensor index notation
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...
applies to any object which has entities that form multidimensional array
In computer science, array is a data type that represents a collection of ''elements'' (values or variables), each selected by one or more indices (identifying keys) that can be computed at run time during program execution. Such a collection i ...
s – not everything with indices is a tensor by default. Instead, tensors are defined by how their coordinates and basis elements change under a transformation from one coordinate system to another.
Note the cross product of two vectors is a pseudovector, while the cross product of a pseudovector with a vector is another vector.
Applications of the tensor and pseudotensor
Other identities can be formed from the tensor and pseudotensor, a notable and very useful identity is one that converts two Levi-Civita symbols adjacently contracted over two indices into an antisymmetrized combination of Kronecker deltas:
The index forms of the dot and cross products, together with this identity, greatly facilitate the manipulation and derivation of other identities in vector calculus and algebra, which in turn are used extensively in physics and engineering. For instance, it is clear the dot and cross products are distributive over vector addition:
without resort to any geometric constructions – the derivation in each case is a quick line of algebra. Although the procedure is less obvious, the vector triple product can also be derived. Rewriting in index notation:
and because cyclic permutations of indices in the symbol does not change its value, cyclically permuting indices in to obtain allows us to use the above - identity to convert the symbols into tensors:
thusly:
Note this is antisymmetric in and , as expected from the left hand side. Similarly, via index notation or even just cyclically relabelling , , and in the previous result and taking the negative:
and the difference in results show that the cross product is not associative. More complex identities, like quadruple products;
and so on, can be derived in a similar manner.
Transformations of Cartesian tensors (any number of dimensions)
Tensors are defined as quantities which transform in a certain way under linear transformations of coordinates.
Second order
Let and be two vectors, so that they transform according to , .
Taking the tensor product gives:
then applying the transformation to the components
and to the bases
gives the transformation law of an order-2 tensor. The tensor is invariant under this transformation:
More generally, for any order-2 tensor
the components transform according to;
and the basis transforms by:
If does not transform according to this rule – whatever quantity may be – it is not an order-2 tensor.
Any order
More generally, for any order tensor
the components transform according to;
and the basis transforms by:
For a pseudotensor
In physics and mathematics, a pseudotensor is usually a quantity that transforms like a tensor under an orientation-preserving coordinate transformation (e.g. a proper rotation) but additionally changes sign under an orientation-reversing coordin ...
of order , the components transform according to;
Pseudovectors as antisymmetric second order tensors
The antisymmetric nature of the cross product can be recast into a tensorial form as follows.[ Let be a vector, be a pseudovector, be another vector, and be a second order tensor such that:
As the cross product is linear in and , the components of can be found by inspection, and they are:
so the pseudovector can be written as an antisymmetric tensor. This transforms as a tensor, not a pseudotensor. For the mechanical example above for the tangential velocity of a rigid body, given by , this can be rewritten as where is the tensor corresponding to the pseudovector :
For an example in ]electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...
, while the electric field
An electric field (sometimes called E-field) is a field (physics), physical field that surrounds electrically charged particles such as electrons. In classical electromagnetism, the electric field of a single charge (or group of charges) descri ...
is a vector field
In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space \mathbb^n. A vector field on a plane can be visualized as a collection of arrows with given magnitudes and dire ...
, the magnetic field
A magnetic field (sometimes called B-field) is a physical field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular ...
is a pseudovector field. These fields are defined from the Lorentz force
In electromagnetism, the Lorentz force is the force exerted on a charged particle by electric and magnetic fields. It determines how charged particles move in electromagnetic environments and underlies many physical phenomena, from the operation ...
for a particle of electric charge
Electric charge (symbol ''q'', sometimes ''Q'') is a physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative''. Like charges repel each other and ...
traveling at velocity :
and considering the second term containing the cross product of a pseudovector and velocity vector , it can be written in matrix form, with , , and as column vectors and as an antisymmetric matrix:
If a pseudovector is explicitly given by a cross product of two vectors (as opposed to entering the cross product with another vector), then such pseudovectors can also be written as antisymmetric tensors of second order, with each entry a component of the cross product. The angular momentum of a classical pointlike particle orbiting about an axis, defined by , is another example of a pseudovector, with corresponding antisymmetric tensor:
Although Cartesian tensors do not occur in the theory of relativity; the tensor form of orbital angular momentum enters the spacelike part of the relativistic angular momentum
In physics, relativistic angular momentum refers to the mathematical formalisms and physical concepts that define angular momentum in special relativity (SR) and general relativity (GR). The relativistic quantity is subtly different from the thre ...
tensor, and the above tensor form of the magnetic field enters the spacelike part of the electromagnetic tensor
In electromagnetism, the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field in spacetime. Th ...
.
Vector and tensor calculus
The following formulae are only so simple in Cartesian coordinates – in general curvilinear coordinates there are factors of the metric and its determinant – see tensors in curvilinear coordinates for more general analysis.
Vector calculus
Following are the differential operators of vector calculus
Vector calculus or vector analysis is a branch of mathematics concerned with the differentiation and integration of vector fields, primarily in three-dimensional Euclidean space, \mathbb^3. The term ''vector calculus'' is sometimes used as a ...
. Throughout, let be a scalar field
In mathematics and physics, a scalar field is a function associating a single number to each point in a region of space – possibly physical space. The scalar may either be a pure mathematical number ( dimensionless) or a scalar physical ...
, and
be vector field
In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space \mathbb^n. A vector field on a plane can be visualized as a collection of arrows with given magnitudes and dire ...
s, in which all scalar and vector fields are functions of the position vector
In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents a point ''P'' in space. Its length represents the distance in relation to an arbitrary reference origin ''O'', and ...
and time .
The gradient
In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. The g ...
operator in Cartesian coordinates is given by:
and in index notation, this is usually abbreviated in various ways:
This operator acts on a scalar field Φ to obtain the vector field directed in the maximum rate of increase of Φ:
The index notation for the dot and cross products carries over to the differential operators of vector calculus.[
The ]directional derivative
In multivariable calculus, the directional derivative measures the rate at which a function changes in a particular direction at a given point.
The directional derivative of a multivariable differentiable (scalar) function along a given vect ...
of a scalar field is the rate of change of along some direction vector (not necessarily a unit vector
In mathematics, a unit vector in a normed vector space is a Vector (mathematics and physics), vector (often a vector (geometry), spatial vector) of Norm (mathematics), length 1. A unit vector is often denoted by a lowercase letter with a circumfle ...
), formed out of the components of and the gradient:
The divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the rate that the vector field alters the volume in an infinitesimal neighborhood of each point. (In 2D this "volume" refers to ...
of a vector field is:
Note the interchange of the components of the gradient and vector field yields a different differential operator
which could act on scalar or vector fields. In fact, if A is replaced by the velocity field
In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the f ...
of a fluid, this is a term in the material derivative
In continuum mechanics, the material derivative describes the time rate of change of some physical quantity (like heat or momentum) of a material element that is subjected to a space-and-time-dependent macroscopic velocity field. The material de ...
(with many other names) of continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the deformation of and transmission of forces through materials modeled as a ''continuous medium'' (also called a ''continuum'') rather than as discrete particles.
Continuum mec ...
, with another term being the partial time derivative:
which usually acts on the velocity field leading to the non-linearity in the Navier-Stokes equations.
As for the curl
cURL (pronounced like "curl", ) is a free and open source computer program for transferring data to and from Internet servers. It can download a URL from a web server over HTTP, and supports a variety of other network protocols, URI scheme ...
of a vector field , this can be defined as a pseudovector field by means of the symbol:
which is only valid in three dimensions, or an antisymmetric tensor field of second order via antisymmetrization of indices, indicated by delimiting the antisymmetrized indices by square brackets (see Ricci calculus Ricci () is an Italian surname. Notable Riccis Arts and entertainment
* Antonio Ricci (painter) (c.1565–c.1635), Spanish Baroque painter of Italian origin
* Christina Ricci (born 1980), American actress
* Clara Ross Ricci (1858-1954), British ...
):
which is valid in any number of dimensions. In each case, the order of the gradient and vector field components should not be interchanged as this would result in a different differential operator:
which could act on scalar or vector fields.
Finally, the Laplacian operator
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the n ...
is defined in two ways, the divergence of the gradient of a scalar field :
or the square of the gradient operator, which acts on a scalar field or a vector field :
In physics and engineering, the gradient, divergence, curl, and Laplacian operator arise inevitably in fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics of fluids (liquids, gases, and plasma (physics), plasmas) and the forces on them.
Originally applied to water (hydromechanics), it found applications in a wide range of discipl ...
, Newtonian gravitation
Newton's law of universal gravitation describes gravity as a force by stating that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the sq ...
, electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...
, heat conduction
Thermal conduction is the diffusion of thermal energy (heat) within one material or between materials in contact. The higher temperature object has molecules with more kinetic energy; collisions between molecules distributes this kinetic energy u ...
, and even quantum mechanics
Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
.
Vector calculus identities can be derived in a similar way to those of vector dot and cross products and combinations. For example, in three dimensions, the curl of a cross product of two vector fields and :
where the product rule
In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v ...
was used, and throughout the differential operator was not interchanged with or . Thus:
Tensor calculus
One can continue the operations on tensors of higher order. Let denote a second order tensor field, again dependent on the position vector and time .
For instance, the gradient of a vector field in two equivalent notations ("dyadic" and "tensor", respectively) is:
which is a tensor field of second order.
The divergence of a tensor is:
which is a vector field. This arises in continuum mechanics in Cauchy's laws of motion – the divergence of the Cauchy stress tensor is a vector field, related to body force
In physics, a body force is a force that acts throughout the volume of a body.Springer site - Book 'Solid mechanics'preview paragraph 'Body forces'./ref> Forces due to gravity, electric fields and magnetic fields are examples of body forces. Bod ...
s acting on the fluid.
Difference from the standard tensor calculus
Cartesian tensors are as in tensor algebra
In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra over a field, algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', ...
, but Euclidean structure of and restriction of the basis brings some simplifications compared to the general theory.
The general tensor algebra consists of general mixed tensor
In tensor analysis, a mixed tensor is a tensor which is neither strictly covariant nor strictly contravariant; at least one of the indices of a mixed tensor will be a subscript (covariant) and at least one of the indices will be a superscript ( ...
s of type :
with basis elements:
the components transform according to:
as for the bases:
For Cartesian tensors, only the order of the tensor matters in a Euclidean space with an orthonormal basis, and all indices can be lowered. A Cartesian basis does not exist unless the vector space has a positive-definite metric, and thus cannot be used in relativistic contexts.
History
Dyadic tensor
In mathematics, specifically multilinear algebra, a dyadic or dyadic tensor is a second order tensor, written in a notation that fits in with vector algebra.
There are numerous ways to multiply two Euclidean vectors. The dot product takes in two ...
s were historically the first approach to formulating second-order tensors, similarly triadic tensors for third-order tensors, and so on. Cartesian tensors use tensor index notation
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...
, in which the variance
In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
may be glossed over and is often ignored, since the components remain unchanged by raising and lowering indices
The asterisk ( ), from Late Latin , from Ancient Greek , , "little star", is a typographical symbol. It is so called because it resembles a conventional image of a heraldic star.
Computer scientists and mathematicians often vocalize it as st ...
.
See also
*Tensor algebra
In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra over a field, algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', ...
*Tensor calculus
In mathematics, a tensor is an algebraic object that describes a multilinear 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 in curvilinear coordinates
*Rotation group
In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...
References
General references
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Further reading and applications
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External links
''Cartesian Tensors''
V. N. Kaliakin, ''Brief Review of Tensors'', University of Delaware
R. E. Hunt, ''Cartesian Tensors'', University of Cambridge
{{Tensors
Linear algebra
Tensors
Applied mathematics