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Cartesian tensorIn geometry and linear algebra, a Cartesian tensoruses an orthonormal basis to represent a tensor in aEuclidean space in the form of components.Converting a tensor's components from one such basisto another is through an orthogonal transformation.

The most familiar coordinate systems are the two-dimensional and three-dimensional Cartesiancoordinate systems. Cartesian tensors may be usedwith any Euclidean space, or more technically, anyfinite-dimensional vector space over the field of realnumbers that has an inner product.

Use of Cartesian tensors occurs in physics andengineering, such as with the Cauchy stress tensorand the moment of inertia tensor in rigid bodydynamics. Sometimes general curvilinear coordinatesare convenient, as in high-deformation continuum mechanics, or even necessary, asin general relativity. While orthonormal bases may be found for some such coordinatesystems (e.g. tangent to spherical coordinates), Cartesian tensors may provideconsiderable simplification for applications in which rotations of rectilinearcoordinate axes suffice. The transformation is a passive transformation, since thecoordinates are changed and not the physical system.

Cartesian basis and related terminology

Vectors in three dimensions

In 3d Euclidean space, �3, the standard basis is ex, ey, ez. Each basis vector points

along the x-, y-, and z-axes, and the vectors are all unit vectors (or normalized), so thebasis is orthonormal.

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Throughout, when referring to Cartesian coordinates in three dimensions, a right-handed system is assumed and this is much more common than a left-handed systemin practice, see orientation (vector space) for details.

For Cartesian tensors of order 1, a Cartesian vector a can be written algebraically as alinear combination of the basis vectors ex, ey, ez:

where the coordinates of the vector with respect to the Cartesian basis are denoted ax,

ay, az. It is common and helpful to display the basis vectors as column vectors

when we have a coordinate vector in a column vector representation:

A row vector representation is also legitimate, although in the context of generalcurvilinear coordinate systems the row and column vector representations are usedseparately for specific reasons – see Einstein notation and covariance andcontravariance of vectors for why.

The term "component" of a vector is ambiguous: it could refer to:

a specific coordinate of the vector such as az (a scalar), and similarly for x and y,

orthe coordinate scalar-multiplying the corresponding basis vector, in which casethe "y-component" of a is ayey (a vector), and similarly for y and z.

A more general notation is tensor index notation, which has the flexibility ofnumerical values rather than fixed coordinate labels. The Cartesian labels are

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replaced by tensor indices in the basis vectors ex � e1, ey � e2, ez � e3 and

coordinates Ax � A1, Ay � A2, Az � A3. In general, the notation e1, e2, e3 refers to any

basis, and A1, A2, A3 refers to the corresponding coordinate system; although here

they are restricted to the Cartesian system. Then:

It is standard to use the Einstein notation – the summation sign for summation overan index repeated only twice within a term may be suppressed for notationalconciseness:

An advantage of the index notation over coordinate-specific notations is theindependence of the dimension of the underlying vector space, i.e. the sameexpression on the right hand side takes the same form in higher dimensions (seebelow). Previously, the Cartesian labels x, y, z were just labels and not indices. (It isinformal 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 ⊗ of twoCartesian vectors a and b, written T = a ⊗ b. Analogous to vectors, it can be writtenas a linear combination of the tensor basis ex ⊗ ex ≡ exx, ex ⊗ ey ≡ exy, ..., ez ⊗ ez ≡ ezz

(the right hand side of each identity is only an abbreviation, nothing more):

Representing each basis tensor as a matrix:

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then T can be represented more systematically as a matrix:

See matrix multiplication for the notational correspondence between matrices andthe dot and tensor products.

More generally, whether or not T is a tensor product of two vectors, it is always alinear combination of the basis tensors with coordinates Txx, Txy, ... Tzz:

while in terms of tensor indices:

and in matrix form:

Second order tensors occur naturally in physics and engineering when physicalquantities have directional dependence in the system, often in a "stimulus-response"way. This can be mathematically seen through one aspect of tensors - they aremultilinear functions. A second order tensor T which takes in a vector u of somemagnitude and direction will return a vector v; of a different magnitude and in adifferent direction to u, in general. The notation used for functions in mathematical

analysis leads us to write v = T(u),[1] while the same idea can be expressed in matrix

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and index notations[2] (including the summation convention), respectively:

By "linear", if u = ρr + σs for two scalars ρ and σ and vectors r and s, then in functionand index notations:

and similarly for the matrix notation. The function, matrix, and index notations allmean 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 acompact manner. Both provide the physical interpretation of directions; vectors haveone direction, while second order tensors connect two directions together. One canassociate a tensor index or coordinate label with a basis vector direction.

The use of second order tensors are the minimum to describe changes in magnitudesand directions of vectors, as the dot product of two vectors is always a scalar, whilethe cross product of two vectors is always a pseudovector perpendicular to the planedefined by the vectors, so these products of vectors alone cannot obtain a new vectorof any magnitude in any direction. (See also below for more on the dot and crossproducts). The tensor product of two vectors is a second order tensor, although thishas no obvious directional interpretation by itself.

The previous idea can be continued: if T takes in two vectors p and q, it will return ascalar r. In function notation we write r = T(p, q), while in matrix and indexnotations (including the summation convention) respectively:

The tensor T is linear in both input vectors. When vectors and tensors are written

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without reference to components, and indices are not used, sometimes a dot · isplaced where summations over indices (known as tensor contractions) are taken. For

the above cases:[1][2]

motivated by the dot product notation:

More generally, a tensor of order m which takes in n vectors (where n is between 0and m inclusive) will return a tensor of order m − n, see Tensor: As multilinear mapsfor further generalizations and details. The concepts above also apply topseudovectors in the same way as for vectors. The vectors and tensors themselves canvary within throughout space, in which case we have vector fields and tensor fields,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 T:

See also constitutive equation for more specialized examples.

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Vectors and tensors in n dimensions

In n-dimensional Euclidean space over the real numbers, �n, the standard basis isdenoted e1, e2, e3, ... en. Each basis vector ei points along the positive xi axis, with the

basis being orthonormal. Component j of ei is given by the Kronecker delta:

A vector in �n takes the form:

Similarly for the order 2 tensor above, for each vector a and b in �n:

or more generally:

Transformations of Cartesian vectors (any number ofdimensions)

Meaning of "invariance" under coordinate transformations

The position vector x in �n is a simple and common example of a vector, and can berepresented in any coordinate system. Consider the case of rectangular coordinatesystems with orthonormal bases only. It is possible to have a coordinate system withrectangular geometry if the basis vectors are all mutually perpendicular and notnormalized, in which case the basis is orthogonal but not orthonormal. However,orthonormal bases are easier to manipulate and are often used in practice. Thefollowing results are true for orthonormal bases, not orthogonal ones.

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In one rectangular coordinate system, x as a

contravector has coordinates xi and basis vectors ei,

while as a covector it has coordinates xi and basis

covectors ei, and we have:

In another rectangular coordinate system, x as a

contravector has coordinates xi and bases ei, while as

a covector it has coordinates xi and bases ei, and we

have:

Each new coordinate is a function of all the old ones, and vice versa for the inversefunction:

and similarly each new basis vector is a function of all the old ones, and vice versa forthe inverse function:

for all i, j.

A vector is invariant under any change of basis, so if coordinates transform accordingto a transformation matrix L, the bases transform according to the matrix inverse L−1, and conversely if the coordinates transform according to inverse L−1, the basestransform according to the matrix L. The difference between each of thesetransformations is shown conventionally through the indices as superscripts for

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contravariance and subscripts for covariance, and the coordinates and bases arelinearly transformed according to the following rules:

where Lij represents the entries of the transformation matrix (row number is i and

column number is j) and (L−1)ik denotes the entries of the inverse matrix of the

matrix Lik.

If L is an orthogonal transformation (orthogonal matrix), the objects transforming byit are defined as Cartesian tensors. This geometrically has the interpretation that arectangular coordinate system is mapped to another rectangular coordinate system,in which the norm of the vector x is preserved (and distances are preserved).

The determinant of L is det(L) = ±1, which corresponds to two types of orthogonaltransformation: (+1) for rotations and (−1) for improper rotations (includingreflections).

There are considerable algebraic simplifications, the matrix transpose is the inversefrom the definition of an orthogonal transformation:

From the previous table, orthogonal transformations of covectors and contravectorsare identical. There is no need to differ between raising and lowering indices, and inthis context and applications to physics and engineering the indices are usually allsubscripted to remove confusion for exponents. All indices will be lowered in theremainder of this article. One can determine the actual raised and lowered indices byconsidering which quantities are covectors or contravectors, and the relevanttransformation rules.

Exactly the same transformation rules apply to any vector a, not only the positionvector. If its components ai do not transform according to the rules, a is not a vector.

Despite the similarity between the expressions above, for the change of coordinates

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such as xj = Lijxi, and the action of a tensor on a vector like bi = Tijaj, L is not a

tensor, but T is. In the change of coordinates, L is a matrix, used to relate tworectangular coordinate systems with orthonormal bases together. For the tensorrelating a vector to a vector, the vectors and tensors throughout the equation allbelong to the same coordinate system and basis.

Derivatives and Jacobian matrix elements

The entries of L are partial derivatives of the new or old coordinates with respect tothe old or new coordinates, respectively.

Differentiating xi with respect to xk:

so

is an element of the Jacobian matrix. There is a (partially mnemonical)correspondence between index positions attached to L and in the partial derivative: iat the top and j at the bottom, in each case, although for Cartesian tensors the indicescan be lowered.

Conversely, differentiating xj with respect to xi:

so

is an element of the inverse Jacobian matrix, with a similar index correspondence.

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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, L depends on the basis chosen. For twoorthonormal bases

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projecting x to the x axes:

projecting x to the x axes:

Hence the components reduce to direction cosines between the xi and xj axes:

where θij and θji are the angles between the xi and xj axes. In general, θij is not equal

to θji, because for example θ12 and θ21 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 xi components equal to the sum of projecting the

xj components onto the xj axes.

The numbers ei⋅ej arranged into a matrix would form a symmetric matrix (a matrix

equal to its own transpose) due to the symmetry in the dot products, in fact it is themetric tensor g. By contrast ei⋅ej or ei⋅ej do not form symmetric matrices in general,

as displayed above. Therefore, while the L matrices are still orthogonal, they are notsymmetric.

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Apart from a rotation about any one axis, in which the xi and xi for some i coincide,

the angles are not the same as Euler angles, and so the L matrices are not the same asthe rotation matrices.

Transformation of the dot and cross products (three dimensionsonly)

The dot product and cross product occur very frequently, in applications of vectoranalysis to physics and engineering, examples include:

power transferred P by an object exerting a force F with velocity v along astraight-line path:

tangential velocity v at a point x of a rotating rigid body with angular velocity ω:

potential energy U of a magnetic dipole of magnetic moment m in a uniformexternal magnetic field B:

angular momentum J for a particle with position vector r and momentum p:

torque τ acting on an electric dipole of electric dipole moment p in a uniformexternal electric field E:

induced surface current density jS in a magnetic material of magnetization M on

a surface with unit normal n:

How these products transform under orthogonal transformations is illustrated below.

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Dot product, Kronecker delta, and metric tensor

The dot product ⋅ of each possible pairing of the basis vectors follows from the basisbeing orthonormal. For perpendicular pairs we have

while for parallel pairs we have

Replacing Cartesian labels by index notation as shown above, these results can besummarized by

where δij are the components of the Kronecker delta. The Cartesian basis can be used

to represent δ in this way.

In addition, each metric tensor component gij 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 δ:

This is not true for general bases: orthogonal coordinates have diagonal metricscontaining various scale factors (i.e. not necessarily 1), while general curvilinearcoordinates could also have nonzero entries for off-diagonal components.

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The dot product of two vectors a and b transforms according to

which is intuitive, since the dot product of two vectors is a single scalar independentof any coordinates. This also applies more generally to any coordinate systems, notjust rectangular ones; the dot product in one coordinate system is the same in anyother.

Cross and product, Levi-Civita symbol, and pseudovectors

Non-zero values of the Levi-Civita symbol εijk as thevolume ei · ej × ek of a cube spanned by the 3dorthonormal basis.

For the cross product × of two vectors, the results are (almost) the other way round.Again, assuming a right-handed 3d Cartesian coordinate system, cyclic permutationsin 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, these can be summarizedby:

where i, j, k are indices which take values 1, 2, 3. It follows that:

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These permutation relations and their corresponding values are important, and thereis an object coinciding with this property: the Levi-Civita symbol, denoted by ε. TheLevi-Civita symbol entries can be represented by the Cartesian basis:

which geometrically corresponds to the volume of a cube spanned by the orthonormalbasis vectors, with sign indicating orientation (and not a "positive or negativevolume"). Here, the orientation is fixed by ε123 = +1, for a right-handed system. A left-

handed system would fix ε123 = −1 or equivalently ε321 = +1.

The scalar triple product can now be written:

with the geometric interpretation of volume (of the parallelepiped spanned by a, b, c)

and algebraically is a determinant:[3]

This in turn can be used to rewrite the cross product of two vectors as follows:

Contrary to its appearance, the Levi-Civita symbol is not a tensor, but apseudotensor, the components transform according to:

Therefore the transformation of the cross product of a and b is:

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and so a × b transforms as a pseudovector, because of the determinant factor.

The tensor index notation applies to any object which has entities that formmultidimensional arrays – not everything with indices is a tensor by default. Instead,tensors are defined by how their coordinates and basis elements change under atransformation from one coordinate system to another.

Note the cross product of two vectors is a pseudovector, while the cross product of apseudovector with a vector is another vector.

Applications of the δ tensor and ε pseudotensor

Other identities can be formed from the δ tensor and ε pseudotensor, a notable andvery useful identity is one that converts two Levi-Civita symbols adjacently contractedover two indices into an antisymmetrized combination of Kronecker deltas:

The index forms of the dot and cross products, together with this identity, greatlyfacilitate the manipulation and derivation of other identities in vector calculus andalgebra, 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 quickline of algebra. Although the procedure is less obvious, the vector triple product canalso be derived. Rewriting in index notation:

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and because cyclic permutations of indices in the ε symbol does not change its value,cyclically permuting indices in εkℓm to obtain εℓmk allows us to use the above δ-ε

identity to convert the ε symbols into δ tensors:

thusly:

Note this is antisymmetric in b and c, as expected from the left hand side. Similarly,via index notation or even just cyclically relabelling a, b, and c in the previous resultand taking the negative:

and the difference in results show that the cross product is not associative. Morecomplex identities, like quadruple products;

and so on, can be derived in a similar manner.

Transformations of Cartesian tensors (any number ofdimensions)

Tensors are defined as quantities which transform in a certain way under lineartransformations of coordinates.

Second order

Let a = aiei and b = biei be two vectors, so that they transform according to aj = aiLij,

bj = biLij.

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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 a⊗b is invariant underthis transformation:

More generally, for any order-2 tensor

the components transform according to;

,

and the basis transforms by:

If R does not transform according to this rule - whatever quantity R may be, it's notan order 2 tensor.

Any order

More generally, for any order p tensor

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the components transform according to;

and the basis transforms by:

For a pseudotensor S of order p, 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.[4] Let c be a vector, a be a pseudovector, b be another vector, and T be asecond order tensor such that:

As the cross product is linear in a and b, the components of T can be found byinspection, and they are:

so the pseudovector a can be written as an antisymmetric tensor. This transforms as atensor, not a pseudotensor. For the mechanical example above for the tangentialvelocity of a rigid body, given by v = ω × x, this can be rewritten as v = Ω · x where Ωis the tensor corresponding to the pseudovector ω:

For an example in electromagnetism, while the electric field E is a vector field, the

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magnetic field B is a pseudovector field. These fields are defined from the Lorentzforce for a particle of electric charge q traveling at velocity v:

and considering the second term containing the cross product of a pseudovector Band velocity vector v, it can be written in matrix form, with F, E, and v as columnvectors and B as an antisymmetric matrix:

If a pseudovector is explicitly given by a cross product of two vectors (as opposed toentering the cross product with another vector), then such pseudovectors can also bewritten as antisymmetric tensors of second order, with each entry a component of thecross product. The angular momentum of a classical pointlike particle orbiting aboutan axis, defined by J = x × p, is another example of a pesudovector, withcorresponding antisymmetric tensor:

Although Cartesian tensors do not occur in the theory of relativity; the tensor form oforbital angular momentum J enters the spacelike part of the relativistic angularmomentum tensor, and the above tensor form of the magnetic field B enters thespacelike part of the electromagnetic tensor.

Vector and tensor calculus

It should be emphasized the following formulae are only so simple in Cartesiancoordinates - in general curvilinear coordinates there are factors of the metric and itsdeterminant - see tensors in curvilinear coordinates for more general analysis.

Vector calculus

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Following are the differential operators of vector calculus. Throughout, left Φ(r, t) bea scalar field, and

be vector fields, in which all scalar and vector fields are functions of the positionvector r and time t.

The gradient 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 themaximum rate of increase of Φ:

The index notation for the dot and cross products carries over to the differential

operators of vector calculus.[5]

The directional derivative of a scalar field Φ is the rate of change of Φ along somedirection vector a (not necessarily a unit vector), formed out of the components of aand the gradient:

The divergence of a vector field A is:

Note the interchange of the components of the gradient and vector field yields a

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different differential operator

which could act on scalar or vector fields. In fact, if A is replaced by the velocity fieldu(r, t) of a fluid, this is a term in the material derivative (with many other names) ofcontinuum mechanics, 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 of a vector field A, this can be defined as a pseudovector field by meansof the ε symbol:

which is only valid in three dimensions, or an antisymmetric tensor field of secondorder via antisymmetrization of indices, indicated by delimiting the antisymmetrizedindices by square brackets (see Ricci calculus):

which is valid in any number of dimensions. In each case, the order of the gradientand vector field components should not be interchanged as this would result in adifferent differential operator:

which could act on scalar or vector fields.

Finally, the Laplacian operator is defined in two ways, the divergence of the gradientof a scalar field Φ:

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or the square of the gradient operator, which acts on a scalar field Φ or a vector fieldA:

In physics and engineering, the gradient, divergence, curl, and Laplacian operatorarise inevitably in fluid mechanics, Newtonian gravitation, electromagnetism, heatconduction, and even quantum mechanics.

Vector calculus identities can be derived in a similar way to those of vector dot andcross products and combinations. For example, in three dimensions, the curl of across product of two vector fields A and B:

where the product rule was used, and throughout the differential operator was notinterchanged with A or B. Thus:

Tensor calculus

One can continue the operations on tensors of higher order. Let T = T(r, t) denote asecond order tensor field, again dependent on the position vector r and time t.

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.

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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 σ is a vector field, related to body forcesacting on the fluid.

Difference from the standard tensor calculus

Cartesian tensors are as in tensor algebra, but Euclidean structure of and restrictionof the basis brings some simplifications compared to the general theory.

The general tensor algebra consists of general mixed tensors of type (p, q):

with basis elements:

the components transform according to:

as for the bases:

For Cartesian tensors, only the order p + q of the tensor matters in a Euclidean spacewith an orthonormal basis, and all p + q indices can be lowered. A Cartesian basisdoes not exist unless the vector space has a positive-definite metric, and thus cannotbe used in relativistic contexts.

History

Dyadic tensors were historically the first approach to formulating second-order