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Cartesian tensor

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In <a href="Geometry" title="Geometry">geometry</a> and <a href="Linear_algebra" title="Linear algebra">linear algebra</a>, a Cartesian tensor uses an <a href="Orthonormal_basis" title="Orthonormal basis">orthonormal basis</a> to <a href="Representation_(mathematics)" title="Representation (mathematics)">represent</a> a <a href="Tensor" title="Tensor">tensor</a> in a <a href="Euclidean_space" title="Euclidean space">Euclidean space</a> in the form of components. Converting a tensor's components from one such basis to another is done through an <a href="Orthogonal_transformation" title="Orthogonal transformation">orthogonal transformation</a>.

The most familiar coordinate systems are the <a href="Plane_(mathematics)" title="Plane (mathematics)">two-dimensional</a> and <a href="Three-dimensional_space" title="Three-dimensional space">three-dimensional</a> <a href="Cartesian_coordinate" class="mw-redirect" title="Cartesian coordinate">Cartesian coordinate</a> systems. Cartesian tensors may be used with any Euclidean space, or more technically, any finite-dimensional <a href="Vector_space" title="Vector space">vector space</a> over the <a href="Field_(mathematics)" title="Field (mathematics)">field</a> of <a href="Real_number" title="Real number">real numbers</a> that has an <a href="Inner_product" class="mw-redirect" title="Inner product">inner product</a>.

Use of Cartesian tensors occurs in <a href="Physics" title="Physics">physics</a> and <a href="Engineering" title="Engineering">engineering</a>, such as with the <a href="Cauchy_stress_tensor" title="Cauchy stress tensor">Cauchy stress tensor</a> and the <a href="Moment_of_inertia" title="Moment of inertia">moment of inertia</a> tensor in <a href="Rigid_body_dynamics" title="Rigid body dynamics">rigid body dynamics</a>. Sometimes general <a href="Curvilinear_coordinates" title="Curvilinear coordinates">curvilinear coordinates</a> are convenient, as in high-deformation <a href="Continuum_mechanics" title="Continuum mechanics">continuum mechanics</a>, or even necessary, as in <a href="General_relativity" title="General relativity">general relativity</a>. While orthonormal bases may be found for some such coordinate systems (e.g. <a href="Tangent" title="Tangent">tangent</a> to <a href="Spherical_coordinate_system" title="Spherical coordinate system">spherical coordinates</a>), Cartesian tensors may provide considerable simplification for applications in which rotations of rectilinear coordinate axes suffice. The transformation is a <a href="Passive_transformation" class="mw-redirect" title="Passive transformation">passive transformation</a>, since the coordinates are changed and not the physical system.

<div class="mw-heading mw-heading2"><h2 id="Cartesian_basis_and_related_terminology">Cartesian basis and related terminology</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Vectors_in_three_dimensions">Vectors in three dimensions</h3></div>

In <a href="Three-dimensional_space" title="Three-dimensional space">3D</a> <a href="Euclidean_space" title="Euclidean space">Euclidean space</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation>

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</math><img src="./f936ddf584f8f3dd2a0ed08917001b7a404c10b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}" loading="lazy">, the <a href="Standard_basis" title="Standard basis">standard basis</a> is ex, ey, ez. Each basis vector points along the x-, y-, and z-axes, and the vectors are all <a href="Unit_vector" title="Unit vector">unit vectors</a> (or normalized), so the basis is <a href="Orthonormal" class="mw-redirect" title="Orthonormal">orthonormal</a>.

Throughout, when referring to <a href="Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">Cartesian coordinates</a> in <a href="Three_dimensions" class="mw-redirect" title="Three dimensions">three dimensions</a>, a right-handed system is assumed and this is much more common than a left-handed system in practice, see <a href="Orientation_(vector_space)" title="Orientation (vector space)">orientation (vector space)</a> for details.

For Cartesian tensors of order 1, a Cartesian vector a can be written algebraically as a <a href="Linear_combination" title="Linear combination">linear combination</a> of the basis vectors ex, ey, ez:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {a} =a_{\text{x}}\mathbf {e} _{\text{x}}+a_{\text{y}}\mathbf {e} _{\text{y}}+a_{\text{z}}\mathbf {e} _{\text{z}}}</annotation>

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where the <a href="Coordinate_vector" title="Coordinate vector">coordinates</a> 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 <a href="Column_vector" class="mw-redirect" title="Column vector">column vectors</a>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {e} _{\text{x}}={\begin{pmatrix}1\\0\\0\end{pmatrix}}\,,\quad \mathbf {e} _{\text{y}}={\begin{pmatrix}0\\1\\0\end{pmatrix}}\,,\quad \mathbf {e} _{\text{z}}={\begin{pmatrix}0\\0\\1\end{pmatrix}}}</annotation>

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when we have a <a href="Coordinate_vector" title="Coordinate vector">coordinate vector</a> in a column vector representation:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {a} ={\begin{pmatrix}a_{\text{x}}\\a_{\text{y}}\\a_{\text{z}}\end{pmatrix}}}</annotation>

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A <a href="Row_vector" class="mw-redirect" title="Row vector">row vector</a> 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 <a href="Einstein_notation" title="Einstein notation">Einstein notation</a> and <a href="Covariance_and_contravariance_of_vectors" title="Covariance and contravariance of vectors">covariance and contravariance of vectors</a> for why.

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

<ul><li>a specific coordinate of the vector such as az (a scalar), and similarly for x and y, or</li>

<li>the coordinate scalar-multiplying the corresponding basis vector, in which case the "y-component" of a is ayey (a vector), and similarly for x and z.</li></ul>

A more general notation is <a href="Tensor_index_notation" class="mw-redirect" title="Tensor index notation">tensor index notation</a>, which has the flexibility of numerical values rather than fixed coordinate labels. The Cartesian labels are 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:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {a} =a_{1}\mathbf {e} _{1}+a_{2}\mathbf {e} _{2}+a_{3}\mathbf {e} _{3}=\sum _{i=1}^{3}a_{i}\mathbf {e} _{i}}</annotation>

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It is standard to use the <a href="Einstein_notation" title="Einstein notation">Einstein notation</a>—the summation sign for summation over an index that is present exactly twice within a term may be suppressed for notational conciseness:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {a} =\sum _{i=1}^{3}a_{i}\mathbf {e} _{i}\equiv a_{i}\mathbf {e} _{i}}</annotation>

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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").

<div class="mw-heading mw-heading3"><h3 id="Second-order_tensors_in_three_dimensions">Second-order tensors in three dimensions</h3></div>

A <a href="Dyadic_tensor" class="mw-redirect" title="Dyadic tensor">dyadic tensor</a> T is an order-2 tensor formed by the <a href="Tensor_product" title="Tensor product">tensor product</a> ⊗ of two Cartesian vectors a and b, written T = a ⊗ b. Analogous to vectors, it can be written as 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):

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {T} =\quad &\left(a_{\text{x}}\mathbf {e} _{\text{x}}+a_{\text{y}}\mathbf {e} _{\text{y}}+a_{\text{z}}\mathbf {e} _{\text{z}}\right)\otimes \left(b_{\text{x}}\mathbf {e} _{\text{x}}+b_{\text{y}}\mathbf {e} _{\text{y}}+b_{\text{z}}\mathbf {e} _{\text{z}}\right)\\[5pt]{}=\quad &a_{\text{x}}b_{\text{x}}\mathbf {e} _{\text{x}}\otimes \mathbf {e} _{\text{x}}+a_{\text{x}}b_{\text{y}}\mathbf {e} _{\text{x}}\otimes \mathbf {e} _{\text{y}}+a_{\text{x}}b_{\text{z}}\mathbf {e} _{\text{x}}\otimes \mathbf {e} _{\text{z}}\\[4pt]{}+{}&a_{\text{y}}b_{\text{x}}\mathbf {e} _{\text{y}}\otimes \mathbf {e} _{\text{x}}+a_{\text{y}}b_{\text{y}}\mathbf {e} _{\text{y}}\otimes \mathbf {e} _{\text{y}}+a_{\text{y}}b_{\text{z}}\mathbf {e} _{\text{y}}\otimes \mathbf {e} _{\text{z}}\\[4pt]{}+{}&a_{\text{z}}b_{\text{x}}\mathbf {e} _{\text{z}}\otimes \mathbf {e} _{\text{x}}+a_{\text{z}}b_{\text{y}}\mathbf {e} _{\text{z}}\otimes \mathbf {e} _{\text{y}}+a_{\text{z}}b_{\text{z}}\mathbf {e} _{\text{z}}\otimes \mathbf {e} _{\text{z}}\end{aligned}}}</annotation>

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Representing each basis tensor as a matrix:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {e} _{\text{x}}\otimes \mathbf {e} _{\text{x}}&\equiv \mathbf {e} _{\text{xx}}={\begin{pmatrix}1&0&0\\0&0&0\\0&0&0\end{pmatrix}}\,,&\mathbf {e} _{\text{x}}\otimes \mathbf {e} _{\text{y}}&\equiv \mathbf {e} _{\text{xy}}={\begin{pmatrix}0&1&0\\0&0&0\\0&0&0\end{pmatrix}}\,,&\mathbf {e} _{\text{z}}\otimes \mathbf {e} _{\text{z}}&\equiv \mathbf {e} _{\text{zz}}={\begin{pmatrix}0&0&0\\0&0&0\\0&0&1\end{pmatrix}}\end{aligned}}}</annotation>

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then T can be represented more systematically as a matrix:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {T} ={\begin{pmatrix}a_{\text{x}}b_{\text{x}}&a_{\text{x}}b_{\text{y}}&a_{\text{x}}b_{\text{z}}\\a_{\text{y}}b_{\text{x}}&a_{\text{y}}b_{\text{y}}&a_{\text{y}}b_{\text{z}}\\a_{\text{z}}b_{\text{x}}&a_{\text{z}}b_{\text{y}}&a_{\text{z}}b_{\text{z}}\end{pmatrix}}}</annotation>

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See <a href="Matrix_multiplication#The_inner_and_outer_products" title="Matrix multiplication">matrix multiplication</a> for the notational correspondence between matrices and the dot and tensor products.

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

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {T} =\quad &T_{\text{xx}}\mathbf {e} _{\text{xx}}+T_{\text{xy}}\mathbf {e} _{\text{xy}}+T_{\text{xz}}\mathbf {e} _{\text{xz}}\\[4pt]{}+{}&T_{\text{yx}}\mathbf {e} _{\text{yx}}+T_{\text{yy}}\mathbf {e} _{\text{yy}}+T_{\text{yz}}\mathbf {e} _{\text{yz}}\\[4pt]{}+{}&T_{\text{zx}}\mathbf {e} _{\text{zx}}+T_{\text{zy}}\mathbf {e} _{\text{zy}}+T_{\text{zz}}\mathbf {e} _{\text{zz}}\end{aligned}}}</annotation>

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while in terms of tensor indices:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {T} =T_{ij}\mathbf {e} _{ij}\equiv \sum _{ij}T_{ij}\mathbf {e} _{i}\otimes \mathbf {e} _{j}\,,}</annotation>

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and in matrix form:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {T} ={\begin{pmatrix}T_{\text{xx}}&T_{\text{xy}}&T_{\text{xz}}\\T_{\text{yx}}&T_{\text{yy}}&T_{\text{yz}}\\T_{\text{zx}}&T_{\text{zy}}&T_{\text{zz}}\end{pmatrix}}}</annotation>

</semantics>

</math>

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 <a href="Multilinear_function" class="mw-redirect" title="Multilinear function">multilinear functions</a>. 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 <a href="Function_(mathematics)" title="Function (mathematics)">functions</a> in <a href="Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a> leads us to write v − T(u),<a href="#cite_note-MTW_notation-1">[1]</a> while the same idea can be expressed in matrix and index notations<a href="#cite_note-Kibble_notation-2">[2]</a> (including the summation convention), respectively:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\begin{pmatrix}v_{\text{x}}\\v_{\text{y}}\\v_{\text{z}}\end{pmatrix}}&={\begin{pmatrix}T_{\text{xx}}&T_{\text{xy}}&T_{\text{xz}}\\T_{\text{yx}}&T_{\text{yy}}&T_{\text{yz}}\\T_{\text{zx}}&T_{\text{zy}}&T_{\text{zz}}\end{pmatrix}}{\begin{pmatrix}u_{\text{x}}\\u_{\text{y}}\\u_{\text{z}}\end{pmatrix}}\,,&v_{i}&=T_{ij}u_{j}\end{aligned}}}</annotation>

</semantics>

</math>

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

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {v} &=&&\mathbf {T} (\rho \mathbf {r} +\sigma \mathbf {s} )&=&&\rho \mathbf {T} (\mathbf {r} )+\sigma \mathbf {T} (\mathbf {s} )\\[1ex]v_{i}&=&&T_{ij}(\rho r_{j}+\sigma s_{j})&=&&\rho T_{ij}r_{j}+\sigma T_{ij}s_{j}\end{aligned}}}</annotation>

</semantics>

</math>

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 <a href="Dot_product" title="Dot product">dot product</a> of two vectors is always a scalar, while the <a href="Cross_product" title="Cross product">cross product</a> 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 T takes in two vectors p and q, it will return a scalar r. In function notation we write r = T(p, q), while in matrix and index notations (including the summation convention) respectively:

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<annotation encoding="application/x-tex">{\displaystyle r={\begin{pmatrix}p_{\text{x}}&p_{\text{y}}&p_{\text{z}}\end{pmatrix}}{\begin{pmatrix}T_{\text{xx}}&T_{\text{xy}}&T_{\text{xz}}\\T_{\text{yx}}&T_{\text{yy}}&T_{\text{yz}}\\T_{\text{zx}}&T_{\text{zy}}&T_{\text{zz}}\end{pmatrix}}{\begin{pmatrix}q_{\text{x}}\\q_{\text{y}}\\q_{\text{z}}\end{pmatrix}}=p_{i}T_{ij}q_{j}}</annotation>

</semantics>

</math>

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 <a href="Tensor_contraction" title="Tensor contraction">tensor contractions</a>) are taken. For the above cases:<a href="#cite_note-MTW_notation-1">[1]</a><a href="#cite_note-Kibble_notation-2">[2]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {v} &=\mathbf {T} \cdot \mathbf {u} \\r&=\mathbf {p} \cdot \mathbf {T} \cdot \mathbf {q} \end{aligned}}}</annotation>

</semantics>

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motivated by the dot product notation:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot \mathbf {b} \equiv a_{i}b_{i}}</annotation>

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</math>

More generally, a tensor of order m which takes in n vectors (where n is between 0 and m inclusive) will return a tensor of order m − n, see <a href="Tensor#As_multilinear_maps" title="Tensor">Tensor § As multilinear maps</a> 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 <a href="Vector_field" title="Vector field">vector fields</a> and <a href="Tensor_field" title="Tensor field">tensor fields</a>, and can also depend on time.

Following are some examples:

<th scope="col">An applied or given...

</th>

<th scope="col">...to a material or object of...

</th>

<th scope="col">...results in...

</th>

<th scope="col">...in the material or object, given by:

</th></tr>

<tr>

<td><a href="Unit_vector" title="Unit vector">unit vector</a> n</td>

<td><a href="Cauchy_stress_tensor" title="Cauchy stress tensor">Cauchy stress tensor</a> σ</td>

<td>a traction force t</td>

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</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {t} ={\boldsymbol {\sigma }}\cdot \mathbf {n} }</annotation>

</semantics>

</math><img src="./0b8ca99e97dae349505896136abe9cb8391bbbcc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.897ex; height:2.009ex;" alt="{\displaystyle \mathbf {t} ={\boldsymbol {\sigma }}\cdot \mathbf {n} }" loading="lazy">

</td></tr>

<tr>

<td scope="row" rowspan="2"><a href="Angular_velocity" title="Angular velocity">angular velocity</a> ω

</td>

<td rowspan="2"><a href="Moment_of_inertia" title="Moment of inertia">moment of inertia</a> I

</td>

<td>an <a href="Angular_momentum" title="Angular momentum">angular momentum</a> J</td>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {J} =\mathbf {I} \cdot {\boldsymbol {\omega }}}</annotation>

</semantics>

</math><img src="./e6deb0fbec84b8c6e8e74c25fb249651326c94ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.841ex; height:2.176ex;" alt="{\displaystyle \mathbf {J} =\mathbf {I} \cdot {\boldsymbol {\omega }}}" loading="lazy">

</td></tr>

<tr>

<td>a rotational <a href="Kinetic_energy" title="Kinetic energy">kinetic energy</a> T</td>

<mfrac>

</mfrac>

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<annotation encoding="application/x-tex">{\displaystyle T={\tfrac {1}{2}}{\boldsymbol {\omega }}\cdot \mathbf {I} \cdot {\boldsymbol {\omega }}}</annotation>

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</math><img src="./b7918eca5ad0fa677225bcfaedf06f7f1bacf09e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.102ex; height:3.509ex;" alt="{\displaystyle T={\tfrac {1}{2}}{\boldsymbol {\omega }}\cdot \mathbf {I} \cdot {\boldsymbol {\omega }}}" loading="lazy">

</td></tr>

<tr>

<td scope="row" rowspan="2"><a href="Electric_field" title="Electric field">electric field</a> E

</td>

<td><a href="Electrical_conductivity" class="mw-redirect" title="Electrical conductivity">electrical conductivity</a> σ</td>

<td>a <a href="Current_density" title="Current density">current density</a> flow J</td>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {J} ={\boldsymbol {\sigma }}\cdot \mathbf {E} }</annotation>

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</math><img src="./929633df3be638db36c3278933f797ec4ac85110.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.51ex; height:2.176ex;" alt="{\displaystyle \mathbf {J} ={\boldsymbol {\sigma }}\cdot \mathbf {E} }" loading="lazy">

</td></tr>

<tr>

<td><a href="Polarizability" title="Polarizability">polarizability</a> α (related to the <a href="Permittivity" title="Permittivity">permittivity</a> ε and <a href="Electric_susceptibility" title="Electric susceptibility">electric susceptibility</a> χE)</td>

<td>an induced <a href="Polarization_density" title="Polarization density">polarization</a> field P</td>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {P} ={\boldsymbol {\alpha }}\cdot \mathbf {E} }</annotation>

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</math><img src="./9d377fe5e4a6401eeb0974e53286146ef4cd6edd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.13ex; height:2.176ex;" alt="{\displaystyle \mathbf {P} ={\boldsymbol {\alpha }}\cdot \mathbf {E} }" loading="lazy">

</td></tr>

<tr>

<td><a href="Magnetic_field" title="Magnetic field">magnetic H field</a></td>

<td><a href="Magnetic_permeability" class="mw-redirect" title="Magnetic permeability">magnetic permeability</a> μ</td>

<td>a <a href="Magnetic_field" title="Magnetic field">magnetic B field</a></td>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {B} ={\boldsymbol {\mu }}\cdot \mathbf {H} }</annotation>

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</math><img src="./70aec864329367c10b04cc31830751cc354e2da2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.416ex; height:2.509ex;" alt="{\displaystyle \mathbf {B} ={\boldsymbol {\mu }}\cdot \mathbf {H} }" loading="lazy">

</td></tr></tbody></table>

For the electrical conduction example, the index and matrix notations would be:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}J_{i}&=\sigma _{ij}E_{j}\equiv \sum _{j}\sigma _{ij}E_{j}\\{\begin{pmatrix}J_{\text{x}}\\J_{\text{y}}\\J_{\text{z}}\end{pmatrix}}&={\begin{pmatrix}\sigma _{\text{xx}}&\sigma _{\text{xy}}&\sigma _{\text{xz}}\\\sigma _{\text{yx}}&\sigma _{\text{yy}}&\sigma _{\text{yz}}\\\sigma _{\text{zx}}&\sigma _{\text{zy}}&\sigma _{\text{zz}}\end{pmatrix}}{\begin{pmatrix}E_{\text{x}}\\E_{\text{y}}\\E_{\text{z}}\end{pmatrix}}\end{aligned}}}</annotation>

</semantics>

</math>

while for the rotational kinetic energy T:

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}T&={\frac {1}{2}}\omega _{i}I_{ij}\omega _{j}\equiv {\frac {1}{2}}\sum _{ij}\omega _{i}I_{ij}\omega _{j}\,,\\&={\frac {1}{2}}{\begin{pmatrix}\omega _{\text{x}}&\omega _{\text{y}}&\omega _{\text{z}}\end{pmatrix}}{\begin{pmatrix}I_{\text{xx}}&I_{\text{xy}}&I_{\text{xz}}\\I_{\text{yx}}&I_{\text{yy}}&I_{\text{yz}}\\I_{\text{zx}}&I_{\text{zy}}&I_{\text{zz}}\end{pmatrix}}{\begin{pmatrix}\omega _{\text{x}}\\\omega _{\text{y}}\\\omega _{\text{z}}\end{pmatrix}}\,.\end{aligned}}}</annotation>

</semantics>

</math>

See also <a href="Constitutive_equation" title="Constitutive equation">constitutive equation</a> for more specialized examples.

<div class="mw-heading mw-heading3"><h3 id="Vectors_and_tensors_in_n_dimensions">Vectors and tensors in n dimensions</h3></div>

In n-dimensional Euclidean space over the real numbers, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation>

</semantics>

</math><img src="./c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy">, the standard basis is denoted 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 <a href="Kronecker_delta" title="Kronecker delta">Kronecker delta</a>:

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<annotation encoding="application/x-tex">{\displaystyle (\mathbf {e} _{i})_{j}=\delta _{ij}}</annotation>

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</math>

A vector in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

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Similarly for the order-2 tensor above, for each vector a and b in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {T} =a_{i}b_{j}\mathbf {e} _{ij}\equiv \sum _{ij}a_{i}b_{j}\mathbf {e} _{i}\otimes \mathbf {e} _{j}\,,}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {T} =T_{ij}\mathbf {e} _{ij}\equiv \sum _{ij}T_{ij}\mathbf {e} _{i}\otimes \mathbf {e} _{j}\,.}</annotation>

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<div class="mw-heading mw-heading2"><h2 id="Transformations_of_Cartesian_vectors_(any_number_of_dimensions)">Transformations of Cartesian vectors (any number of dimensions)</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Meaning_of_"invariance"_under_coordinate_transformations">Meaning of "invariance" under coordinate transformations</h3></div>

The <a href="Position_vector" class="mw-redirect" title="Position vector">position vector</a> x in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

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</math><img src="./c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"> is a simple and common example of a vector, and can be represented in any <a href="Coordinate_system" title="Coordinate system">coordinate system</a>. Consider the case of <a href="Rectangular_coordinate_system" class="mw-redirect" title="Rectangular coordinate system">rectangular coordinate systems</a> 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 orthogonal but not orthonormal. 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, 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:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {x} &=x^{i}\mathbf {e} _{i}\,,&\mathbf {x} &=x_{i}\mathbf {e} ^{i}\end{aligned}}}</annotation>

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In another rectangular coordinate system, x as a contravector has coordinates xi and basis ei, while as a covector it has coordinates xi and basis ei, and we have:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {x} &={\bar {x}}^{i}{\bar {\mathbf {e} }}_{i}\,,&\mathbf {x} &={\bar {x}}_{i}{\bar {\mathbf {e} }}^{i}\end{aligned}}}</annotation>

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Each new coordinate is a function of all the old ones, and vice versa for the <a href="Inverse_function" title="Inverse function">inverse function</a>:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\bar {x}}{}^{i}={\bar {x}}{}^{i}\left(x^{1},x^{2},\ldots \right)\quad &\rightleftharpoons \quad x{}^{i}=x{}^{i}\left({\bar {x}}^{1},{\bar {x}}^{2},\ldots \right)\\{\bar {x}}{}_{i}={\bar {x}}{}_{i}\left(x_{1},x_{2},\ldots \right)\quad &\rightleftharpoons \quad x{}_{i}=x{}_{i}\left({\bar {x}}_{1},{\bar {x}}_{2},\ldots \right)\end{aligned}}}</annotation>

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

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\bar {\mathbf {e} }}{}_{j}={\bar {\mathbf {e} }}{}_{j}\left(\mathbf {e} _{1},\mathbf {e} _{2},\ldots \right)\quad &\rightleftharpoons \quad \mathbf {e} {}_{j}=\mathbf {e} {}_{j}\left({\bar {\mathbf {e} }}_{1},{\bar {\mathbf {e} }}_{2},\ldots \right)\\{\bar {\mathbf {e} }}{}^{j}={\bar {\mathbf {e} }}{}^{j}\left(\mathbf {e} ^{1},\mathbf {e} ^{2},\ldots \right)\quad &\rightleftharpoons \quad \mathbf {e} {}^{j}=\mathbf {e} {}^{j}\left({\bar {\mathbf {e} }}^{1},{\bar {\mathbf {e} }}^{2},\ldots \right)\end{aligned}}}</annotation>

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for all i, j.

A vector is invariant under any <a href="Change_of_basis" title="Change of basis">change of basis</a>, so if coordinates transform according to a <a href="Transformation_matrix" title="Transformation matrix">transformation matrix</a> L, the bases transform according to the <a href="Matrix_inverse" class="mw-redirect" title="Matrix inverse">matrix inverse</a> L−1, and conversely if the coordinates transform according to inverse L−1, the bases transform according to the matrix L. 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 <a href="Linear_transformation" class="mw-redirect" title="Linear transformation">linearly transformed</a> according to the following rules:

<th>Vector elements

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<th>Contravariant transformation law

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<th>Covariant transformation law

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<th>Coordinates

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<annotation encoding="application/x-tex">{\displaystyle {\bar {x}}^{j}=x^{i}({\boldsymbol {\mathsf {L}}})_{i}{}^{j}=x^{i}{\mathsf {L}}_{i}{}^{j}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\bar {x}}_{j}=x_{k}\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{j}{}^{k}}</annotation>

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</math><img src="./7c5ffe21a470f5537aaa97e8e2fc2ade167dfe18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:15.568ex; height:3.843ex;" alt="{\displaystyle {\bar {x}}_{j}=x_{k}\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{j}{}^{k}}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle {\bar {\mathbf {e} }}_{j}=\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{j}{}^{k}\mathbf {e} _{k}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\bar {\mathbf {e} }}^{j}=({\boldsymbol {\mathsf {L}}})_{i}{}^{j}\mathbf {e} ^{i}={\mathsf {L}}_{i}{}^{j}\mathbf {e} ^{i}}</annotation>

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</math><img src="./aa1376b10b9bc139217792b6f3cc5569a095ec0a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.22ex; height:3.176ex;" alt="{\displaystyle {\bar {\mathbf {e} }}^{j}=({\boldsymbol {\mathsf {L}}})_{i}{}^{j}\mathbf {e} ^{i}={\mathsf {L}}_{i}{}^{j}\mathbf {e} ^{i}}" loading="lazy">

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<th>Any vector

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<annotation encoding="application/x-tex">{\displaystyle {\bar {x}}^{j}{\bar {\mathbf {e} }}_{j}=x^{i}{\mathsf {L}}_{i}{}^{j}\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{j}{}^{k}\mathbf {e} _{k}=x^{i}\delta _{i}{}^{k}\mathbf {e} _{k}=x^{i}\mathbf {e} _{i}}</annotation>

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</math><img src="./131d20ca0b0ba9eeba99e593e3aaa4da7590a5e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:40.412ex; height:3.843ex;" alt="{\displaystyle {\bar {x}}^{j}{\bar {\mathbf {e} }}_{j}=x^{i}{\mathsf {L}}_{i}{}^{j}\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{j}{}^{k}\mathbf {e} _{k}=x^{i}\delta _{i}{}^{k}\mathbf {e} _{k}=x^{i}\mathbf {e} _{i}}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle {\bar {x}}_{j}{\bar {\mathbf {e} }}^{j}=x_{i}\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{j}{}^{i}{\mathsf {L}}_{k}{}^{j}\mathbf {e} ^{k}=x_{i}\delta ^{i}{}_{k}\mathbf {e} ^{k}=x_{i}\mathbf {e} ^{i}}</annotation>

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</math><img src="./87312f685794555112f488458032c5e1dc76b649.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:40.433ex; height:3.843ex;" alt="{\displaystyle {\bar {x}}_{j}{\bar {\mathbf {e} }}^{j}=x_{i}\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{j}{}^{i}{\mathsf {L}}_{k}{}^{j}\mathbf {e} ^{k}=x_{i}\delta ^{i}{}_{k}\mathbf {e} ^{k}=x_{i}\mathbf {e} ^{i}}" loading="lazy">

</td></tr>

</tbody></table>

where Lij represents the entries of the <a href="Transformation_matrix" title="Transformation matrix">transformation matrix</a> (row number is i and column number is j) and (L−1)ik denotes the entries of the <a href="Inverse_matrix" class="mw-redirect" title="Inverse matrix">inverse matrix</a> of the matrix Lik.

If L is an <a href="Orthogonal_transformation" title="Orthogonal transformation">orthogonal transformation</a> (<a href="Orthogonal_matrix" title="Orthogonal matrix">orthogonal matrix</a>), 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 <a href="Norm_(mathematics)" title="Norm (mathematics)">norm</a> of the vector x is preserved (and distances are preserved).

The <a href="Determinant" title="Determinant">determinant</a> of L is det(L) = ±1, which corresponds to two types of orthogonal transformation: (+1) for <a href="Rotation_(mathematics)" title="Rotation (mathematics)">rotations</a> and (−1) for <a href="Improper_rotation" title="Improper rotation">improper rotations</a> (including <a href="Reflection_(mathematics)" title="Reflection (mathematics)">reflections</a>).

There are considerable algebraic simplifications, the <a href="Matrix_transpose" class="mw-redirect" title="Matrix transpose">matrix transpose</a> is the <a href="Inverse_matrix" class="mw-redirect" title="Inverse matrix">inverse</a> from the definition of an orthogonal transformation:

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<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\mathsf {L}}}^{\textsf {T}}={\boldsymbol {\mathsf {L}}}^{-1}\Rightarrow \left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{i}{}^{j}=\left({\boldsymbol {\mathsf {L}}}^{\textsf {T}}\right)_{i}{}^{j}=({\boldsymbol {\mathsf {L}}})^{j}{}_{i}={\mathsf {L}}^{j}{}_{i}}</annotation>

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From the previous table, orthogonal transformations of covectors and contravectors are identical. There is no need to differ between <a href="Raising_and_lowering_indices" class="mw-redirect" title="Raising and lowering indices">raising and lowering indices</a>, and in this context and applications to physics and engineering the indices are usually all subscripted to remove confusion for <a href="Exponent" class="mw-redirect" title="Exponent">exponents</a>. 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 a, not only the position vector. 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 such as xj = Lijxi, and the action of a tensor on a vector like bi = Tij aj, L is not a tensor, but T is. In the change of coordinates, L 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.

<div class="mw-heading mw-heading3"><h3 id="Derivatives_and_Jacobian_matrix_elements">Derivatives and Jacobian matrix elements</h3></div>

The entries of L are <a href="Partial_derivative" title="Partial derivative">partial derivatives</a> of the new or old coordinates with respect to the old or new coordinates, respectively.

Differentiating xi with respect to xk:

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial {\bar {x}}_{i}}{\partial x_{k}}}={\frac {\partial }{\partial x_{k}}}(x_{j}{\mathsf {L}}_{ji})={\mathsf {L}}_{ji}{\frac {\partial x_{j}}{\partial x_{k}}}=\delta _{kj}{\mathsf {L}}_{ji}={\mathsf {L}}_{ki}}</annotation>

</semantics>

</math>

so

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<annotation encoding="application/x-tex">{\displaystyle {{\mathsf {L}}_{i}}^{j}\equiv {\mathsf {L}}_{ij}={\frac {\partial {\bar {x}}_{j}}{\partial x_{i}}}}</annotation>

</semantics>

</math>

is an element of the <a href="Jacobian_matrix" class="mw-redirect" title="Jacobian matrix">Jacobian matrix</a>. 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 xj with respect to xi:

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial x_{j}}{\partial {\bar {x}}_{k}}}={\frac {\partial }{\partial {\bar {x}}_{k}}}\left({\bar {x}}_{i}\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{ij}\right)={\frac {\partial {\bar {x}}_{i}}{\partial {\bar {x}}_{k}}}\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{ij}=\delta _{ki}\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{ij}=\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{kj}}</annotation>

</semantics>

</math>

so

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<annotation encoding="application/x-tex">{\displaystyle \left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{i}{}^{j}\equiv \left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{ij}={\frac {\partial x_{j}}{\partial {\bar {x}}_{i}}}}</annotation>

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

Many sources state transformations in terms of the partial derivatives:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{c}\displaystyle {\bar {x}}_{j}=x_{i}{\frac {\partial {\bar {x}}_{j}}{\partial x_{i}}}\\[3pt]\upharpoonleft \downharpoonright \\[3pt]\displaystyle x_{j}={\bar {x}}_{i}{\frac {\partial x_{j}}{\partial {\bar {x}}_{i}}}\end{array}}}</annotation>

</semantics>

</math><img src="./8b21de707dc5fe75b7114edd3cbd484967cbfe43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.171ex; width:12.612ex; height:17.343ex;" alt="{\displaystyle {\begin{array}{c}\displaystyle {\bar {x}}_{j}=x_{i}{\frac {\partial {\bar {x}}_{j}}{\partial x_{i}}}\\[3pt]\upharpoonleft \downharpoonright \\[3pt]\displaystyle x_{j}={\bar {x}}_{i}{\frac {\partial x_{j}}{\partial {\bar {x}}_{i}}}\end{array}}}" loading="lazy">

</div>

and the explicit matrix equations in 3d are:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\bar {\mathbf {x} }}&={\boldsymbol {\mathsf {L}}}\mathbf {x} \\{\begin{pmatrix}{\bar {x}}_{1}\\{\bar {x}}_{2}\\{\bar {x}}_{3}\end{pmatrix}}&={\begin{pmatrix}{\frac {\partial {\bar {x}}_{1}}{\partial x_{1}}}&{\frac {\partial {\bar {x}}_{1}}{\partial x_{2}}}&{\frac {\partial {\bar {x}}_{1}}{\partial x_{3}}}\\{\frac {\partial {\bar {x}}_{2}}{\partial x_{1}}}&{\frac {\partial {\bar {x}}_{2}}{\partial x_{2}}}&{\frac {\partial {\bar {x}}_{2}}{\partial x_{3}}}\\{\frac {\partial {\bar {x}}_{3}}{\partial x_{1}}}&{\frac {\partial {\bar {x}}_{3}}{\partial x_{2}}}&{\frac {\partial {\bar {x}}_{3}}{\partial x_{3}}}\end{pmatrix}}{\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}\end{aligned}}}</annotation>

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</math>

similarly for

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} ={\boldsymbol {\mathsf {L}}}^{-1}{\bar {\mathbf {x} }}={\boldsymbol {\mathsf {L}}}^{\textsf {T}}{\bar {\mathbf {x} }}}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading3"><h3 id="Projections_along_coordinate_axes">Projections along coordinate axes</h3></div>

As with all linear transformations, L depends on the basis chosen. For two orthonormal bases

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\bar {\mathbf {e} }}_{i}\cdot {\bar {\mathbf {e} }}_{j}&=\mathbf {e} _{i}\cdot \mathbf {e} _{j}=\delta _{ij}\,,&\left|\mathbf {e} _{i}\right|&=\left|{\bar {\mathbf {e} }}_{i}\right|=1\,,\end{aligned}}}</annotation>

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<ul><li>projecting x to the x axes: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {x}}_{i}={\bar {\mathbf {e} }}_{i}\cdot \mathbf {x} ={\bar {\mathbf {e} }}_{i}\cdot x_{j}\mathbf {e} _{j}=x_{i}{\mathsf {L}}_{ij}\,,}">

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<annotation encoding="application/x-tex">{\displaystyle {\bar {x}}_{i}={\bar {\mathbf {e} }}_{i}\cdot \mathbf {x} ={\bar {\mathbf {e} }}_{i}\cdot x_{j}\mathbf {e} _{j}=x_{i}{\mathsf {L}}_{ij}\,,}</annotation>

</semantics>

</math><img src="./eb6ad8ab826099a8b4372428fc2dac8d20f31e7e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.518ex; height:2.843ex;" alt="{\displaystyle {\bar {x}}_{i}={\bar {\mathbf {e} }}_{i}\cdot \mathbf {x} ={\bar {\mathbf {e} }}_{i}\cdot x_{j}\mathbf {e} _{j}=x_{i}{\mathsf {L}}_{ij}\,,}" loading="lazy"></li>

<li>projecting x to the x axes: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}=\mathbf {e} _{i}\cdot \mathbf {x} =\mathbf {e} _{i}\cdot {\bar {x}}_{j}{\bar {\mathbf {e} }}_{j}={\bar {x}}_{j}\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{ji}\,.}">

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<annotation encoding="application/x-tex">{\displaystyle x_{i}=\mathbf {e} _{i}\cdot \mathbf {x} =\mathbf {e} _{i}\cdot {\bar {x}}_{j}{\bar {\mathbf {e} }}_{j}={\bar {x}}_{j}\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{ji}\,.}</annotation>

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</math><img src="./9a693849c038d6fa254ebcbd9536db3385151461.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:35.182ex; height:3.843ex;" alt="{\displaystyle x_{i}=\mathbf {e} _{i}\cdot \mathbf {x} =\mathbf {e} _{i}\cdot {\bar {x}}_{j}{\bar {\mathbf {e} }}_{j}={\bar {x}}_{j}\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{ji}\,.}" loading="lazy"></li></ul>

Hence the components reduce to <a href="Direction_cosine" title="Direction cosine">direction cosines</a> between the xi and xj axes:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\mathsf {L}}_{ij}&={\bar {\mathbf {e} }}_{i}\cdot \mathbf {e} _{j}=\cos \theta _{ij}\\\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{ij}&=\mathbf {e} _{i}\cdot {\bar {\mathbf {e} }}_{j}=\cos \theta _{ji}\end{aligned}}}</annotation>

</semantics>

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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:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{c}{\bar {x}}_{j}=x_{i}\left({\bar {\mathbf {e} }}_{i}\cdot \mathbf {e} _{j}\right)=x_{i}\cos \theta _{ij}\\[3pt]\upharpoonleft \downharpoonright \\[3pt]x_{j}={\bar {x}}_{i}\left(\mathbf {e} _{i}\cdot {\bar {\mathbf {e} }}_{j}\right)={\bar {x}}_{i}\cos \theta _{ji}\end{array}}}</annotation>

</semantics>

</math><img src="./75af0615bd7dbb450be3a3d806f724fe3e9c2cd7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:27.935ex; height:11.176ex;" alt="{\displaystyle {\begin{array}{c}{\bar {x}}_{j}=x_{i}\left({\bar {\mathbf {e} }}_{i}\cdot \mathbf {e} _{j}\right)=x_{i}\cos \theta _{ij}\\[3pt]\upharpoonleft \downharpoonright \\[3pt]x_{j}={\bar {x}}_{i}\left(\mathbf {e} _{i}\cdot {\bar {\mathbf {e} }}_{j}\right)={\bar {x}}_{i}\cos \theta _{ji}\end{array}}}" loading="lazy">

</div>

and the explicit matrix equations in 3d are:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\bar {\mathbf {x} }}&={\boldsymbol {\mathsf {L}}}\mathbf {x} \\{\begin{pmatrix}{\bar {x}}_{1}\\{\bar {x}}_{2}\\{\bar {x}}_{3}\end{pmatrix}}&={\begin{pmatrix}{\bar {\mathbf {e} }}_{1}\cdot \mathbf {e} _{1}&{\bar {\mathbf {e} }}_{1}\cdot \mathbf {e} _{2}&{\bar {\mathbf {e} }}_{1}\cdot \mathbf {e} _{3}\\{\bar {\mathbf {e} }}_{2}\cdot \mathbf {e} _{1}&{\bar {\mathbf {e} }}_{2}\cdot \mathbf {e} _{2}&{\bar {\mathbf {e} }}_{2}\cdot \mathbf {e} _{3}\\{\bar {\mathbf {e} }}_{3}\cdot \mathbf {e} _{1}&{\bar {\mathbf {e} }}_{3}\cdot \mathbf {e} _{2}&{\bar {\mathbf {e} }}_{3}\cdot \mathbf {e} _{3}\end{pmatrix}}{\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}={\begin{pmatrix}\cos \theta _{11}&\cos \theta _{12}&\cos \theta _{13}\\\cos \theta _{21}&\cos \theta _{22}&\cos \theta _{23}\\\cos \theta _{31}&\cos \theta _{32}&\cos \theta _{33}\end{pmatrix}}{\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}\end{aligned}}}</annotation>

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similarly for

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} ={\boldsymbol {\mathsf {L}}}^{-1}{\bar {\mathbf {x} }}={\boldsymbol {\mathsf {L}}}^{\textsf {T}}{\bar {\mathbf {x} }}}</annotation>

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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 <a href="Symmetric_matrix" title="Symmetric matrix">symmetric matrix</a> (a matrix equal to its own transpose) due to the symmetry in the dot products, in fact it is the <a href="Metric_tensor" title="Metric tensor">metric tensor</a> 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 not symmetric.

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 <a href="Euler_angle" class="mw-redirect" title="Euler angle">Euler angles</a>, and so the L matrices are not the same as the <a href="Rotation_matrix" title="Rotation matrix">rotation matrices</a>.

<div class="mw-heading mw-heading2"><h2 id="Transformation_of_the_dot_and_cross_products_(three_dimensions_only)">Transformation of the dot and cross products (three dimensions only)</h2></div>

The <a href="Dot_product" title="Dot product">dot product</a> and <a href="Cross_product" title="Cross product">cross product</a> occur very frequently, in applications of vector analysis to physics and engineering, examples include:

<ul><li><a href="Power_(physics)" title="Power (physics)">power</a> transferred P by an object exerting a force F with velocity v along a straight-line path: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=\mathbf {v} \cdot \mathbf {F} }">

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<annotation encoding="application/x-tex">{\displaystyle P=\mathbf {v} \cdot \mathbf {F} }</annotation>

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<li>tangential <a href="Velocity" title="Velocity">velocity</a> v at a point x of a rotating <a href="Rigid_body" title="Rigid body">rigid body</a> with <a href="Angular_velocity" title="Angular velocity">angular velocity</a> ω: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} ={\boldsymbol {\omega }}\times \mathbf {x} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {v} ={\boldsymbol {\omega }}\times \mathbf {x} }</annotation>

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<li><a href="Potential_energy" title="Potential energy">potential energy</a> U of a <a href="Magnetic_dipole" title="Magnetic dipole">magnetic dipole</a> of <a href="Magnetic_moment" title="Magnetic moment">magnetic moment</a> m in a uniform external <a href="Magnetic_field" title="Magnetic field">magnetic field</a> B: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U=-\mathbf {m} \cdot \mathbf {B} }">

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<annotation encoding="application/x-tex">{\displaystyle U=-\mathbf {m} \cdot \mathbf {B} }</annotation>

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<li><a href="Angular_momentum" title="Angular momentum">angular momentum</a> J for a particle with <a href="Position_vector" class="mw-redirect" title="Position vector">position vector</a> r and <a href="Momentum" title="Momentum">momentum</a> p: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {J} =\mathbf {r} \times \mathbf {p} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {J} =\mathbf {r} \times \mathbf {p} }</annotation>

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<li><a href="Torque" title="Torque">torque</a> τ acting on an <a href="Electric_dipole" class="mw-redirect" title="Electric dipole">electric dipole</a> of <a href="Electric_dipole_moment" title="Electric dipole moment">electric dipole moment</a> p in a uniform external <a href="Electric_field" title="Electric field">electric field</a> E: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\tau }}=\mathbf {p} \times \mathbf {E} }">

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<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\tau }}=\mathbf {p} \times \mathbf {E} }</annotation>

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<li>induced surface <a href="Current_density" title="Current density">current density</a> jS in a magnetic material of <a href="Magnetization" title="Magnetization">magnetization</a> M on a surface with <a href="Unit_normal" class="mw-redirect" title="Unit normal">unit normal</a> n: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {j} _{\mathrm {S} }=\mathbf {M} \times \mathbf {n} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {j} _{\mathrm {S} }=\mathbf {M} \times \mathbf {n} }</annotation>

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How these products transform under orthogonal transformations is illustrated below.

<div class="mw-heading mw-heading3"><h3 id="Dot_product,_Kronecker_delta,_and_metric_tensor">Dot product, Kronecker delta, and metric tensor</h3></div>

The <a href="Dot_product" title="Dot product">dot product</a> ⋅ of each possible pairing of the basis vectors follows from the basis being orthonormal. For perpendicular pairs we have

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<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{llll}\mathbf {e} _{\text{x}}\cdot \mathbf {e} _{\text{y}}&=\mathbf {e} _{\text{y}}\cdot \mathbf {e} _{\text{z}}&=\mathbf {e} _{\text{z}}\cdot \mathbf {e} _{\text{x}}&=\\\mathbf {e} _{\text{y}}\cdot \mathbf {e} _{\text{x}}&=\mathbf {e} _{\text{z}}\cdot \mathbf {e} _{\text{y}}&=\mathbf {e} _{\text{x}}\cdot \mathbf {e} _{\text{z}}&=0\end{array}}}</annotation>

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while for parallel pairs we have

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {e} _{\text{x}}\cdot \mathbf {e} _{\text{x}}=\mathbf {e} _{\text{y}}\cdot \mathbf {e} _{\text{y}}=\mathbf {e} _{\text{z}}\cdot \mathbf {e} _{\text{z}}=1.}</annotation>

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Replacing Cartesian labels by index notation as shown <a href="#indices_replace_labels">above</a>, these results can be summarized by

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {e} _{i}\cdot \mathbf {e} _{j}=\delta _{ij}}</annotation>

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where δij are the components of the <a href="Kronecker_delta" title="Kronecker delta">Kronecker delta</a>. The Cartesian basis can be used to represent δ in this way.

In addition, each <a href="Metric_tensor" title="Metric tensor">metric tensor</a> component gij with respect to any basis is the dot product of a pairing of basis vectors:

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<annotation encoding="application/x-tex">{\displaystyle g_{ij}=\mathbf {e} _{i}\cdot \mathbf {e} _{j}.}</annotation>

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For the Cartesian basis the components arranged into a matrix are:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {g} ={\begin{pmatrix}g_{\text{xx}}&g_{\text{xy}}&g_{\text{xz}}\\g_{\text{yx}}&g_{\text{yy}}&g_{\text{yz}}\\g_{\text{zx}}&g_{\text{zy}}&g_{\text{zz}}\\\end{pmatrix}}={\begin{pmatrix}\mathbf {e} _{\text{x}}\cdot \mathbf {e} _{\text{x}}&\mathbf {e} _{\text{x}}\cdot \mathbf {e} _{\text{y}}&\mathbf {e} _{\text{x}}\cdot \mathbf {e} _{\text{z}}\\\mathbf {e} _{\text{y}}\cdot \mathbf {e} _{\text{x}}&\mathbf {e} _{\text{y}}\cdot \mathbf {e} _{\text{y}}&\mathbf {e} _{\text{y}}\cdot \mathbf {e} _{\text{z}}\\\mathbf {e} _{\text{z}}\cdot \mathbf {e} _{\text{x}}&\mathbf {e} _{\text{z}}\cdot \mathbf {e} _{\text{y}}&\mathbf {e} _{\text{z}}\cdot \mathbf {e} _{\text{z}}\\\end{pmatrix}}={\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\\\end{pmatrix}}}</annotation>

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so are the simplest possible for the metric tensor, namely the <a href="Identity_matrix" title="Identity matrix">δ</a>:

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This is not true for general bases: <a href="Orthogonal_coordinates" title="Orthogonal coordinates">orthogonal coordinates</a> have <a href="Diagonal_matrix" title="Diagonal matrix">diagonal</a> metrics containing various scale factors (i.e. not necessarily 1), while general <a href="Curvilinear_coordinates" title="Curvilinear coordinates">curvilinear coordinates</a> could also have nonzero entries for off-diagonal components.

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

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot \mathbf {b} ={\bar {a}}_{j}{\bar {b}}_{j}=a_{i}{\mathsf {L}}_{ij}b_{k}\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{jk}=a_{i}\delta _{i}{}_{k}b_{k}=a_{i}b_{i}}</annotation>

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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.

<div class="mw-heading mw-heading3"><h3 id="Cross_product,_Levi-Civita_symbol,_and_pseudovectors">Cross product, Levi-Civita symbol, and pseudovectors</h3></div>

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</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:332px;max-width:332px"><div class="trow"><div class="tsingle" style="width:153px;max-width:153px"><div class="thumbimage" style="height:199px;overflow:hidden"></div><div class="thumbcaption">Cyclic permutations of index values and positively oriented cubic volume.</div></div><div class="tsingle" style="width:175px;max-width:175px"><div class="thumbimage" style="height:199px;overflow:hidden"></div><div class="thumbcaption">Anticyclic permutations of index values and negatively oriented cubic volume.</div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">Non-zero values of the <a href="Levi-Civita_symbol" title="Levi-Civita symbol">Levi-Civita symbol</a> εijk as the volume ei ⋅ ej × ek of a cube spanned by the 3d orthonormal basis.</div></div></div></div>

For the <a href="Cross_product" title="Cross product">cross product</a> (×) of two vectors, the results are (almost) the other way round. Again, assuming a right-handed 3d Cartesian coordinate system, <a href="Cyclic_permutation" title="Cyclic permutation">cyclic permutations</a> in perpendicular directions yield the next vector in the cyclic collection of vectors:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {e} _{\text{x}}\times \mathbf {e} _{\text{y}}&=\mathbf {e} _{\text{z}}&\mathbf {e} _{\text{y}}\times \mathbf {e} _{\text{z}}&=\mathbf {e} _{\text{x}}&\mathbf {e} _{\text{z}}\times \mathbf {e} _{\text{x}}&=\mathbf {e} _{\text{y}}\\[1ex]\mathbf {e} _{\text{y}}\times \mathbf {e} _{\text{x}}&=-\mathbf {e} _{\text{z}}&\mathbf {e} _{\text{z}}\times \mathbf {e} _{\text{y}}&=-\mathbf {e} _{\text{x}}&\mathbf {e} _{\text{x}}\times \mathbf {e} _{\text{z}}&=-\mathbf {e} _{\text{y}}\end{aligned}}}</annotation>

</semantics>

</math>

while parallel vectors clearly vanish:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {e} _{\text{x}}\times \mathbf {e} _{\text{x}}=\mathbf {e} _{\text{y}}\times \mathbf {e} _{\text{y}}=\mathbf {e} _{\text{z}}\times \mathbf {e} _{\text{z}}={\boldsymbol {0}}}</annotation>

</semantics>

</math>

and replacing Cartesian labels by index notation as <a href="#indices_replace_labels">above</a>, these can be summarized by:

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<mtext>cyclic permutations: </mtext>

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<mtd>

<mtext>anticyclic permutations: </mtext>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {e} _{i}\times \mathbf {e} _{j}={\begin{cases}+\mathbf {e} _{k}&{\text{cyclic permutations: }}(i,j,k)=(1,2,3),(2,3,1),(3,1,2)\\[2pt]-\mathbf {e} _{k}&{\text{anticyclic permutations: }}(i,j,k)=(2,1,3),(3,2,1),(1,3,2)\\[2pt]{\boldsymbol {0}}&i=j\end{cases}}}</annotation>

</semantics>

</math>

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

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<mtext>cyclic permutations: </mtext>

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<mtext>anticyclic permutations: </mtext>

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<annotation encoding="application/x-tex">{\displaystyle {\mathbf {e} _{k}\cdot \mathbf {e} _{i}\times \mathbf {e} _{j}}={\begin{cases}+1&{\text{cyclic permutations: }}(i,j,k)=(1,2,3),(2,3,1),(3,1,2)\\[2pt]-1&{\text{anticyclic permutations: }}(i,j,k)=(2,1,3),(3,2,1),(1,3,2)\\[2pt]0&i=j{\text{ or }}j=k{\text{ or }}k=i\end{cases}}}</annotation>

</semantics>

</math>

These permutation relations and their corresponding values are important, and there is an object coinciding with this property: the <a href="Levi-Civita_symbol" title="Levi-Civita symbol">Levi-Civita symbol</a>, denoted by ε. The Levi-Civita symbol entries can be represented by the Cartesian basis:

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<annotation encoding="application/x-tex">{\displaystyle \varepsilon _{ijk}=\mathbf {e} _{i}\cdot \mathbf {e} _{j}\times \mathbf {e} _{k}}</annotation>

</semantics>

</math>

which geometrically corresponds to the <a href="Volume" title="Volume">volume</a> of a <a href="Cube" title="Cube">cube</a> spanned by the orthonormal basis vectors, with sign indicating <a href="Orientation_(vector_space)" title="Orientation (vector space)">orientation</a> (and not a "positive or negative volume"). 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 <a href="Scalar_triple_product" class="mw-redirect" title="Scalar triple product">scalar triple product</a> can now be written:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {c} \cdot \mathbf {a} \times \mathbf {b} =c_{i}\mathbf {e} _{i}\cdot a_{j}\mathbf {e} _{j}\times b_{k}\mathbf {e} _{k}=\varepsilon _{ijk}c_{i}a_{j}b_{k}}</annotation>

</semantics>

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with the geometric interpretation of volume (of the <a href="Parallelepiped" title="Parallelepiped">parallelepiped</a> spanned by a, b, c) and algebraically is a <a href="Determinant" title="Determinant">determinant</a>:<a href="#cite_note-Spiegel-3">[3]</a>: 23

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {c} \cdot \mathbf {a} \times \mathbf {b} ={\begin{vmatrix}c_{\text{x}}&a_{\text{x}}&b_{\text{x}}\\c_{\text{y}}&a_{\text{y}}&b_{\text{y}}\\c_{\text{z}}&a_{\text{z}}&b_{\text{z}}\end{vmatrix}}}</annotation>

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This in turn can be used to rewrite the <a href="Cross_product" title="Cross product">cross product</a> of two vectors as follows:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(\mathbf {a} \times \mathbf {b} )_{i}={\mathbf {e} _{i}\cdot \mathbf {a} \times \mathbf {b} }&=\varepsilon _{\ell jk}{(\mathbf {e} _{i})}_{\ell }a_{j}b_{k}=\varepsilon _{\ell jk}\delta _{i\ell }a_{j}b_{k}=\varepsilon _{ijk}a_{j}b_{k}\\\Rightarrow \quad {\mathbf {a} \times \mathbf {b} }=(\mathbf {a} \times \mathbf {b} )_{i}\mathbf {e} _{i}&=\varepsilon _{ijk}a_{j}b_{k}\mathbf {e} _{i}\end{aligned}}}</annotation>

</semantics>

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Contrary to its appearance, the Levi-Civita symbol is not a tensor, but a <a href="Pseudotensor" title="Pseudotensor">pseudotensor</a>, the components transform according to:

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<annotation encoding="application/x-tex">{\displaystyle {\bar {\varepsilon }}_{pqr}=\det({\boldsymbol {\mathsf {L}}})\varepsilon _{ijk}{\mathsf {L}}_{ip}{\mathsf {L}}_{jq}{\mathsf {L}}_{kr}\,.}</annotation>

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

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&\left({\bar {\mathbf {a} }}\times {\bar {\mathbf {b} }}\right)_{i}\\[1ex]{}={}&{\bar {\varepsilon }}_{ijk}{\bar {a}}_{j}{\bar {b}}_{k}\\[1ex]{}={}&\det({\boldsymbol {\mathsf {L}}})\;\;\varepsilon _{pqr}\;\;{\mathsf {L}}_{pi}{\mathsf {L}}_{qj}{\mathsf {L}}_{rk}\;\;a_{m}{\mathsf {L}}_{mj}\;\;b_{n}{\mathsf {L}}_{nk}\\[1ex]{}={}&\det({\boldsymbol {\mathsf {L}}})\;\;\varepsilon _{pqr}\;\;{\mathsf {L}}_{pi}\;\;{\mathsf {L}}_{qj}\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{jm}\;\;{\mathsf {L}}_{rk}\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{kn}\;\;a_{m}\;\;b_{n}\\[1ex]{}={}&\det({\boldsymbol {\mathsf {L}}})\;\;\varepsilon _{pqr}\;\;{\mathsf {L}}_{pi}\;\;\delta _{qm}\;\;\delta _{rn}\;\;a_{m}\;\;b_{n}\\[1ex]{}={}&\det({\boldsymbol {\mathsf {L}}})\;\;{\mathsf {L}}_{pi}\;\;\varepsilon _{pqr}a_{q}b_{r}\\[1ex]{}={}&\det({\boldsymbol {\mathsf {L}}})\;\;(\mathbf {a} \times \mathbf {b} )_{p}{\mathsf {L}}_{pi}\end{aligned}}}</annotation>

</semantics>

</math>

and so a × b transforms as a <a href="Pseudovector" title="Pseudovector">pseudovector</a>, because of the determinant factor.

The <a href="Tensor_index_notation" class="mw-redirect" title="Tensor index notation">tensor index notation</a> applies to any object which has entities that form <a href="Multidimensional_array" class="mw-redirect" title="Multidimensional array">multidimensional arrays</a> – 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.

<div class="mw-heading mw-heading3"><h3 id="Applications_of_the_δ_tensor_and_ε_pseudotensor">Applications of the δ tensor and ε pseudotensor</h3></div>

Other identities can be formed from the δ tensor and ε 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:

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<annotation encoding="application/x-tex">{\displaystyle \varepsilon _{ijk}\varepsilon _{pqk}=\delta _{ip}\delta _{jq}-\delta _{iq}\delta _{jp}}</annotation>

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</math>

The index forms of the dot and cross products, together with this identity, greatly facilitate the manipulation and derivation of other <a href="Vector_calculus_identities" title="Vector calculus identities">identities in vector calculus</a> 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:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {a} \cdot (\mathbf {b} +\mathbf {c} )&=a_{i}(b_{i}+c_{i})=a_{i}b_{i}+a_{i}c_{i}=\mathbf {a} \cdot \mathbf {b} +\mathbf {a} \cdot \mathbf {c} \\[1ex]\mathbf {a} \times (\mathbf {b} +\mathbf {c} )&=\mathbf {e} _{i}\varepsilon _{ijk}a_{j}(b_{k}+c_{k})=\mathbf {e} _{i}\varepsilon _{ijk}a_{j}b_{k}+\mathbf {e} _{i}\varepsilon _{ijk}a_{j}c_{k}=\mathbf {a} \times \mathbf {b} +\mathbf {a} \times \mathbf {c} \end{aligned}}}</annotation>

</semantics>

</math>

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:

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<annotation encoding="application/x-tex">{\displaystyle \left[\mathbf {a} \times (\mathbf {b} \times \mathbf {c} )\right]_{i}=\varepsilon _{ijk}a_{j}(\varepsilon _{k\ell m}b_{\ell }c_{m})=(\varepsilon _{ijk}\varepsilon _{k\ell m})a_{j}b_{\ell }c_{m}}</annotation>

</semantics>

</math>

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:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\left[\mathbf {a} \times (\mathbf {b} \times \mathbf {c} )\right]_{i}{}={}&\left(\delta _{i\ell }\delta _{jm}-\delta _{im}\delta _{j\ell }\right)a_{j}b_{\ell }c_{m}\\{}={}&\delta _{i\ell }\delta _{jm}a_{j}b_{\ell }c_{m}-\delta _{im}\delta _{j\ell }a_{j}b_{\ell }c_{m}\\{}={}&a_{j}b_{i}c_{j}-a_{j}b_{j}c_{i}\\{}={}&\left[(\mathbf {a} \cdot \mathbf {c} )\mathbf {b} -(\mathbf {a} \cdot \mathbf {b} )\mathbf {c} \right]_{i}\end{aligned}}}</annotation>

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thusly:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \times (\mathbf {b} \times \mathbf {c} )=(\mathbf {a} \cdot \mathbf {c} )\mathbf {b} -(\mathbf {a} \cdot \mathbf {b} )\mathbf {c} }</annotation>

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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 result and taking the negative:

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<annotation encoding="application/x-tex">{\displaystyle (\mathbf {a} \times \mathbf {b} )\times \mathbf {c} =(\mathbf {c} \cdot \mathbf {a} )\mathbf {b} -(\mathbf {c} \cdot \mathbf {b} )\mathbf {a} }</annotation>

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

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<annotation encoding="application/x-tex">{\displaystyle (\mathbf {a} \times \mathbf {b} )\cdot (\mathbf {c} \times \mathbf {d} ),\quad (\mathbf {a} \times \mathbf {b} )\times (\mathbf {c} \times \mathbf {d} ),\quad \ldots }</annotation>

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

<div class="mw-heading mw-heading2"><h2 id="Transformations_of_Cartesian_tensors_(any_number_of_dimensions)">Transformations of Cartesian tensors (any number of dimensions)</h2></div>

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

<div class="mw-heading mw-heading3"><h3 id="Second_order">Second order</h3></div>

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

Taking the tensor product gives:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \otimes \mathbf {b} =a_{i}\mathbf {e} _{i}\otimes b_{j}\mathbf {e} _{j}=a_{i}b_{j}\mathbf {e} _{i}\otimes \mathbf {e} _{j}}</annotation>

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then applying the transformation to the components

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<annotation encoding="application/x-tex">{\displaystyle {\bar {a}}_{p}{\bar {b}}_{q}=a_{i}{\mathsf {L}}_{i}{}_{p}b_{j}{\mathsf {L}}_{j}{}_{q}={\mathsf {L}}_{i}{}_{p}{\mathsf {L}}_{j}{}_{q}a_{i}b_{j}}</annotation>

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and to the bases

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<annotation encoding="application/x-tex">{\displaystyle {\bar {\mathbf {e} }}_{p}\otimes {\bar {\mathbf {e} }}_{q}=\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{pi}\mathbf {e} _{i}\otimes \left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{qj}\mathbf {e} _{j}=\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{pi}\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{qj}\mathbf {e} _{i}\otimes \mathbf {e} _{j}={\mathsf {L}}_{ip}^{-1}{\mathsf {L}}_{jq}^{-1}\mathbf {e} _{i}\otimes \mathbf {e} _{j}}</annotation>

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gives the transformation law of an order-2 tensor. The tensor a⊗b is invariant under this transformation:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\bar {a}}_{p}{\bar {b}}_{q}{\bar {\mathbf {e} }}_{p}\otimes {\bar {\mathbf {e} }}_{q}{}={}&{\mathsf {L}}_{kp}{\mathsf {L}}_{\ell q}a_{k}b_{\ell }\,\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{pi}\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{qj}\mathbf {e} _{i}\otimes \mathbf {e} _{j}\\[1ex]{}={}&{\mathsf {L}}_{kp}\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{pi}{\mathsf {L}}_{\ell q}\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{qj}\,a_{k}b_{\ell }\mathbf {e} _{i}\otimes \mathbf {e} _{j}\\[1ex]{}={}&\delta _{k}{}_{i}\delta _{\ell j}\,a_{k}b_{\ell }\mathbf {e} _{i}\otimes \mathbf {e} _{j}\\[1ex]{}={}&a_{i}b_{j}\mathbf {e} _{i}\otimes \mathbf {e} _{j}\end{aligned}}}</annotation>

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More generally, for any order-2 tensor

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {R} =R_{ij}\mathbf {e} _{i}\otimes \mathbf {e} _{j}\,,}</annotation>

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the components transform according to;

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<annotation encoding="application/x-tex">{\displaystyle {\bar {R}}_{pq}={\mathsf {L}}_{i}{}_{p}{\mathsf {L}}_{j}{}_{q}R_{ij},}</annotation>

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and the basis transforms by:

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If R does not transform according to this rule – whatever quantity R may be – it is not an order-2 tensor.

<div class="mw-heading mw-heading3"><h3 id="Any_order">Any order</h3></div>

More generally, for any order p tensor

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {T} =T_{j_{1}j_{2}\cdots j_{p}}\mathbf {e} _{j_{1}}\otimes \mathbf {e} _{j_{2}}\otimes \cdots \mathbf {e} _{j_{p}}}</annotation>

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the components transform according to;

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<annotation encoding="application/x-tex">{\displaystyle {\bar {T}}_{j_{1}j_{2}\cdots j_{p}}={\mathsf {L}}_{i_{1}j_{1}}{\mathsf {L}}_{i_{2}j_{2}}\cdots {\mathsf {L}}_{i_{p}j_{p}}T_{i_{1}i_{2}\cdots i_{p}}}</annotation>

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and the basis transforms by:

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<annotation encoding="application/x-tex">{\displaystyle {\bar {\mathbf {e} }}_{j_{1}}\otimes {\bar {\mathbf {e} }}_{j_{2}}\cdots \otimes {\bar {\mathbf {e} }}_{j_{p}}=\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{j_{1}i_{1}}\mathbf {e} _{i_{1}}\otimes \left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{j_{2}i_{2}}\mathbf {e} _{i_{2}}\cdots \otimes \left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{j_{p}i_{p}}\mathbf {e} _{i_{p}}}</annotation>

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For a <a href="Pseudotensor" title="Pseudotensor">pseudotensor</a> S of order p, the components transform according to;

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<annotation encoding="application/x-tex">{\displaystyle {\bar {S}}_{j_{1}j_{2}\cdots j_{p}}=\det({\boldsymbol {\mathsf {L}}}){\mathsf {L}}_{i_{1}j_{1}}{\mathsf {L}}_{i_{2}j_{2}}\cdots {\mathsf {L}}_{i_{p}j_{p}}S_{i_{1}i_{2}\cdots i_{p}}\,.}</annotation>

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<div class="mw-heading mw-heading2"><h2 id="Pseudovectors_as_antisymmetric_second_order_tensors">Pseudovectors as antisymmetric second order tensors</h2></div>

The antisymmetric nature of the cross product can be recast into a tensorial form as follows.<a href="#cite_note-Kibble_notation-2">[2]</a> Let c be a vector, a be a pseudovector, b be another vector, and T be a second order tensor such that:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {c} =\mathbf {a} \times \mathbf {b} =\mathbf {T} \cdot \mathbf {b} }</annotation>

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

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {T} ={\begin{pmatrix}0&-a_{\text{z}}&a_{\text{y}}\\a_{\text{z}}&0&-a_{\text{x}}\\-a_{\text{y}}&a_{\text{x}}&0\\\end{pmatrix}}}</annotation>

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so the pseudovector a can be written as an <a href="Antisymmetric_tensor" title="Antisymmetric tensor">antisymmetric tensor</a>. This transforms as a tensor, not a pseudotensor. For the mechanical example above for the tangential velocity of a rigid body, given by v = ω × x, this can be rewritten as v = Ω ⋅ x where Ω is the tensor corresponding to the pseudovector ω:

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<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\Omega }}={\begin{pmatrix}0&-\omega _{\text{z}}&\omega _{\text{y}}\\\omega _{\text{z}}&0&-\omega _{\text{x}}\\-\omega _{\text{y}}&\omega _{\text{x}}&0\\\end{pmatrix}}}</annotation>

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For an example in <a href="Electromagnetism" title="Electromagnetism">electromagnetism</a>, while the <a href="Electric_field" title="Electric field">electric field</a> E is a <a href="Vector_field" title="Vector field">vector field</a>, the <a href="Magnetic_field" title="Magnetic field">magnetic field</a> B is a pseudovector field. These fields are defined from the <a href="Lorentz_force" title="Lorentz force">Lorentz force</a> for a particle of <a href="Electric_charge" title="Electric charge">electric charge</a> q traveling at velocity v:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =q(\mathbf {E} +\mathbf {v} \times \mathbf {B} )=q(\mathbf {E} -\mathbf {B} \times \mathbf {v} )}</annotation>

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

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<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}F_{\text{x}}\\F_{\text{y}}\\F_{\text{z}}\\\end{pmatrix}}=q{\begin{pmatrix}E_{\text{x}}\\E_{\text{y}}\\E_{\text{z}}\\\end{pmatrix}}-q{\begin{pmatrix}0&-B_{\text{z}}&B_{\text{y}}\\B_{\text{z}}&0&-B_{\text{x}}\\-B_{\text{y}}&B_{\text{x}}&0\\\end{pmatrix}}{\begin{pmatrix}v_{\text{x}}\\v_{\text{y}}\\v_{\text{z}}\\\end{pmatrix}}}</annotation>

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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 J = x × p, is another example of a pseudovector, with corresponding antisymmetric tensor:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {J} ={\begin{pmatrix}0&-J_{\text{z}}&J_{\text{y}}\\J_{\text{z}}&0&-J_{\text{x}}\\-J_{\text{y}}&J_{\text{x}}&0\\\end{pmatrix}}={\begin{pmatrix}0&-(xp_{\text{y}}-yp_{\text{x}})&(zp_{\text{x}}-xp_{\text{z}})\\(xp_{\text{y}}-yp_{\text{x}})&0&-(yp_{\text{z}}-zp_{\text{y}})\\-(zp_{\text{x}}-xp_{\text{z}})&(yp_{\text{z}}-zp_{\text{y}})&0\\\end{pmatrix}}}</annotation>

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Although Cartesian tensors do not occur in the theory of relativity; the tensor form of orbital angular momentum J enters the spacelike part of the <a href="Relativistic_angular_momentum" title="Relativistic angular momentum">relativistic angular momentum</a> tensor, and the above tensor form of the magnetic field B enters the spacelike part of the <a href="Electromagnetic_tensor" title="Electromagnetic tensor">electromagnetic tensor</a>.

<div class="mw-heading mw-heading2"><h2 id="Vector_and_tensor_calculus">Vector and tensor calculus</h2></div>

The following formulae are only so simple in Cartesian coordinates – in general curvilinear coordinates there are factors of the metric and its determinant – see <a href="Tensors_in_curvilinear_coordinates" title="Tensors in curvilinear coordinates">tensors in curvilinear coordinates</a> for more general analysis.

<div class="mw-heading mw-heading3"><h3 id="Vector_calculus">Vector calculus</h3></div>

Following are the differential operators of <a href="Vector_calculus" title="Vector calculus">vector calculus</a>. Throughout, let Φ(r, t) be a <a href="Scalar_field" title="Scalar field">scalar field</a>, and

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {A} (\mathbf {r} ,t)&=A_{\text{x}}(\mathbf {r} ,t)\mathbf {e} _{\text{x}}+A_{\text{y}}(\mathbf {r} ,t)\mathbf {e} _{\text{y}}+A_{\text{z}}(\mathbf {r} ,t)\mathbf {e} _{\text{z}}\\[1ex]\mathbf {B} (\mathbf {r} ,t)&=B_{\text{x}}(\mathbf {r} ,t)\mathbf {e} _{\text{x}}+B_{\text{y}}(\mathbf {r} ,t)\mathbf {e} _{\text{y}}+B_{\text{z}}(\mathbf {r} ,t)\mathbf {e} _{\text{z}}\end{aligned}}}</annotation>

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be <a href="Vector_field" title="Vector field">vector fields</a>, in which all scalar and vector fields are functions of the <a href="Position_vector" class="mw-redirect" title="Position vector">position vector</a> r and time t.

The <a href="Gradient" title="Gradient">gradient</a> operator in Cartesian coordinates is given by:

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<annotation encoding="application/x-tex">{\displaystyle \nabla =\mathbf {e} _{\text{x}}{\frac {\partial }{\partial x}}+\mathbf {e} _{\text{y}}{\frac {\partial }{\partial y}}+\mathbf {e} _{\text{z}}{\frac {\partial }{\partial z}}}</annotation>

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and in index notation, this is usually abbreviated in various ways:

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<annotation encoding="application/x-tex">{\displaystyle \nabla _{i}\equiv \partial _{i}\equiv {\frac {\partial }{\partial x_{i}}}}</annotation>

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This operator acts on a scalar field Φ to obtain the vector field directed in the maximum rate of increase of Φ:

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<annotation encoding="application/x-tex">{\displaystyle \left(\nabla \Phi \right)_{i}=\nabla _{i}\Phi }</annotation>

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The index notation for the dot and cross products carries over to the differential operators of vector calculus.<a href="#cite_note-Spiegel-3">[3]</a>: 197

The <a href="Directional_derivative" title="Directional derivative">directional derivative</a> of a scalar field Φ is the rate of change of Φ along some direction vector a (not necessarily a <a href="Unit_vector" title="Unit vector">unit vector</a>), formed out of the components of a and the gradient:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot (\nabla \Phi )=a_{j}(\nabla \Phi )_{j}}</annotation>

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The <a href="Divergence" title="Divergence">divergence</a> of a vector field A is:

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<annotation encoding="application/x-tex">{\displaystyle \nabla \cdot \mathbf {A} =\nabla _{i}A_{i}}</annotation>

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Note the interchange of the components of the gradient and vector field yields a different differential operator

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {A} \cdot \nabla =A_{i}\nabla _{i}}</annotation>

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which could act on scalar or vector fields. In fact, if A is replaced by the <a href="Velocity_field" class="mw-redirect" title="Velocity field">velocity field</a> u(r, t) of a fluid, this is a term in the <a href="Material_derivative" title="Material derivative">material derivative</a> (with many other names) of <a href="Continuum_mechanics" title="Continuum mechanics">continuum mechanics</a>, with another term being the partial <a href="Time_derivative" title="Time derivative">time derivative</a>:

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<annotation encoding="application/x-tex">{\displaystyle {\frac {D}{Dt}}={\frac {\partial }{\partial t}}+\mathbf {u} \cdot \nabla }</annotation>

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which usually acts on the velocity field leading to the non-linearity in the <a href="Navier-Stokes_equations" class="mw-redirect" title="Navier-Stokes equations">Navier-Stokes equations</a>.

As for the <a href="Curl_(mathematics)" title="Curl (mathematics)">curl</a> of a vector field A, this can be defined as a pseudovector field by means of the ε symbol:

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<annotation encoding="application/x-tex">{\displaystyle \left(\nabla \times \mathbf {A} \right)_{i}=\varepsilon _{ijk}\nabla _{j}A_{k}}</annotation>

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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 <a href="Ricci_calculus" title="Ricci calculus">Ricci calculus</a>):

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<annotation encoding="application/x-tex">{\displaystyle \left(\nabla \times \mathbf {A} \right)_{ij}=\nabla _{i}A_{j}-\nabla _{j}A_{i}=2\nabla _{[i}A_{j]}}</annotation>

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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:

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<annotation encoding="application/x-tex">{\displaystyle \varepsilon _{ijk}A_{j}\nabla _{k}=A_{i}\nabla _{j}-A_{j}\nabla _{i}=2A_{[i}\nabla _{j]}}</annotation>

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which could act on scalar or vector fields.

Finally, the <a href="Laplacian_operator" class="mw-redirect" title="Laplacian operator">Laplacian operator</a> is defined in two ways, the divergence of the gradient of 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 field A:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(\nabla \cdot \nabla )\Phi &=(\nabla _{i}\nabla _{i})\Phi \\(\nabla \cdot \nabla )\mathbf {A} &=(\nabla _{i}\nabla _{i})\mathbf {A} \end{aligned}}}</annotation>

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In physics and engineering, the gradient, divergence, curl, and Laplacian operator arise inevitably in <a href="Fluid_mechanics" title="Fluid mechanics">fluid mechanics</a>, <a href="Newtonian_gravitation" class="mw-redirect" title="Newtonian gravitation">Newtonian gravitation</a>, <a href="Electromagnetism" title="Electromagnetism">electromagnetism</a>, <a href="Heat_conduction" class="mw-redirect" title="Heat conduction">heat conduction</a>, and even <a href="Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>.

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 A and B:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&\left[\nabla \times (\mathbf {A} \times \mathbf {B} )\right]_{i}\\{}={}&\varepsilon _{ijk}\nabla _{j}(\varepsilon _{k\ell m}A_{\ell }B_{m})\\{}={}&(\varepsilon _{ijk}\varepsilon _{\ell mk})\nabla _{j}(A_{\ell }B_{m})\\{}={}&(\delta _{i\ell }\delta _{jm}-\delta _{im}\delta _{j\ell })(B_{m}\nabla _{j}A_{\ell }+A_{\ell }\nabla _{j}B_{m})\\{}={}&(B_{j}\nabla _{j}A_{i}+A_{i}\nabla _{j}B_{j})-(B_{i}\nabla _{j}A_{j}+A_{j}\nabla _{j}B_{i})\\{}={}&(B_{j}\nabla _{j})A_{i}+A_{i}(\nabla _{j}B_{j})-B_{i}(\nabla _{j}A_{j})-(A_{j}\nabla _{j})B_{i}\\{}={}&\left[(\mathbf {B} \cdot \nabla )\mathbf {A} +\mathbf {A} (\nabla \cdot \mathbf {B} )-\mathbf {B} (\nabla \cdot \mathbf {A} )-(\mathbf {A} \cdot \nabla )\mathbf {B} \right]_{i}\\\end{aligned}}}</annotation>

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where the <a href="Product_rule" title="Product rule">product rule</a> was used, and throughout the differential operator was not interchanged with A or B. Thus:

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<annotation encoding="application/x-tex">{\displaystyle \nabla \times (\mathbf {A} \times \mathbf {B} )=(\mathbf {B} \cdot \nabla )\mathbf {A} +\mathbf {A} (\nabla \cdot \mathbf {B} )-\mathbf {B} (\nabla \cdot \mathbf {A} )-(\mathbf {A} \cdot \nabla )\mathbf {B} }</annotation>

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<div class="mw-heading mw-heading3"><h3 id="Tensor_calculus">Tensor calculus</h3></div>

One can continue the operations on tensors of higher order. Let T = T(r, t) denote a second 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:

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which is a tensor field of second order.

The divergence of a tensor is:

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which is a vector field. This arises in continuum mechanics in <a href="Continuum_mechanics#Governing_equations" title="Continuum mechanics">Cauchy's laws of motion</a> – the divergence of the Cauchy stress tensor σ is a vector field, related to <a href="Body_force" title="Body force">body forces</a> acting on the fluid.

<div class="mw-heading mw-heading2"><h2 id="Difference_from_the_standard_tensor_calculus">Difference from the standard tensor calculus</h2></div>

Cartesian tensors are as in <a href="Tensor_algebra" title="Tensor algebra">tensor algebra</a>, but Euclidean structure of and restriction of the basis brings some simplifications compared to the general theory.

The general tensor algebra consists of general <a href="Mixed_tensor" title="Mixed tensor">mixed tensors</a> of type (p, q):

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {T} =T_{j_{1}j_{2}\cdots j_{q}}^{i_{1}i_{2}\cdots i_{p}}\mathbf {e} _{i_{1}i_{2}\cdots i_{p}}^{j_{1}j_{2}\cdots j_{q}}}</annotation>

</semantics>

</math>

with basis elements:

</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {e} _{i_{1}i_{2}\cdots i_{p}}^{j_{1}j_{2}\cdots j_{q}}=\mathbf {e} _{i_{1}}\otimes \mathbf {e} _{i_{2}}\otimes \cdots \mathbf {e} _{i_{p}}\otimes \mathbf {e} ^{j_{1}}\otimes \mathbf {e} ^{j_{2}}\otimes \cdots \mathbf {e} ^{j_{q}}}</annotation>

</semantics>

</math>

the components transform according to:

<mover>

</mover>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

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<mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\bar {T}}_{\ell _{1}\ell _{2}\cdots \ell _{q}}^{k_{1}k_{2}\cdots k_{p}}={\mathsf {L}}_{i_{1}}{}^{k_{1}}{\mathsf {L}}_{i_{2}}{}^{k_{2}}\cdots {\mathsf {L}}_{i_{p}}{}^{k_{p}}\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{\ell _{1}}{}^{j_{1}}\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{\ell _{2}}{}^{j_{2}}\cdots \left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{\ell _{q}}{}^{j_{q}}T_{j_{1}j_{2}\cdots j_{q}}^{i_{1}i_{2}\cdots i_{p}}}</annotation>

</semantics>

</math>

as for the bases:

<mover>

</mrow>

</mover>

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</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\bar {\mathbf {e} }}_{k_{1}k_{2}\cdots k_{p}}^{\ell _{1}\ell _{2}\cdots \ell _{q}}=\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{k_{1}}{}^{i_{1}}\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{k_{2}}{}^{i_{2}}\cdots \left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{k_{p}}{}^{i_{p}}{\mathsf {L}}_{j_{1}}{}^{\ell _{1}}{\mathsf {L}}_{j_{2}}{}^{\ell _{2}}\cdots {\mathsf {L}}_{j_{q}}{}^{\ell _{q}}\mathbf {e} _{i_{1}i_{2}\cdots i_{p}}^{j_{1}j_{2}\cdots j_{q}}}</annotation>

</semantics>

</math>

For Cartesian tensors, only the order p + q of the tensor matters in a Euclidean space with an orthonormal basis, and all p + q 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 <a href="Special_relativity" title="Special relativity">relativistic</a> contexts.

<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

<a href="Dyadic_tensor" class="mw-redirect" title="Dyadic tensor">Dyadic tensors</a> were historically the first approach to formulating second-order tensors, similarly triadic tensors for third-order tensors, and so on. Cartesian tensors use <a href="Tensor_index_notation" class="mw-redirect" title="Tensor index notation">tensor index notation</a>, in which the <a href="Covariance_and_contravariance_of_vectors" title="Covariance and contravariance of vectors">variance</a> may be glossed over and is often ignored, since the components remain unchanged by <a href="Raising_and_lowering_indices" class="mw-redirect" title="Raising and lowering indices">raising and lowering indices</a>.

<ul><li><a href="Tensor_algebra" title="Tensor algebra">Tensor algebra</a></li>

<li><a href="Tensor_calculus" class="mw-redirect" title="Tensor calculus">Tensor calculus</a></li>

<li><a href="Tensors_in_curvilinear_coordinates" title="Tensors in curvilinear coordinates">Tensors in curvilinear coordinates</a></li>

<li><a href="Rotation_group" class="mw-redirect" title="Rotation group">Rotation group</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

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</style><cite id="CITEREFC.W._MisnerK.S._ThorneJ.A._Wheeler1973" class="citation book cs1"><a href="Charles_W._Misner" title="Charles W. Misner">C.W. Misner</a>; <a href="Kip_S._Thorne" class="mw-redirect" title="Kip S. Thorne">K.S. Thorne</a>; <a href="John_A._Wheeler" class="mw-redirect" title="John A. Wheeler">J.A. Wheeler</a> (15 September 1973). <a href="Gravitation_(book)" title="Gravitation (book)">Gravitation</a>. Macmillan. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-7167-0344-0</bdi>.</cite>, used throughout

</li>

<li id="cite_note-Kibble_notation-2">^ <a href="#cite_ref-Kibble_notation_2-0">a</a> <a href="#cite_ref-Kibble_notation_2-1">b</a> <a href="#cite_ref-Kibble_notation_2-2">c</a> <cite id="CITEREFT._W._B._Kibble1973" class="citation book cs1">T. W. B. Kibble (1973). Classical Mechanics. European physics series (2nd ed.). McGraw Hill. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-07-084018-8</bdi>.</cite>, see Appendix C.

</li>

<li id="cite_note-Spiegel-3">^ <a href="#cite_ref-Spiegel_3-0">a</a> <a href="#cite_ref-Spiegel_3-1">b</a> <cite id="CITEREFM._R._SpiegelS._LipcshutzD._Spellman2009" class="citation book cs1">M. R. Spiegel; S. Lipcshutz; D. Spellman (2009). Vector analysis. Schaum's Outlines (2nd ed.). McGraw Hill. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-07-161545-7</bdi>.</cite>

</li>

</ol></div></div>

<div class="mw-heading mw-heading3"><h3 id="General_references">General references</h3></div>

<ul><li><cite id="CITEREFD._C._Kay1988" class="citation book cs1">D. C. Kay (1988). Tensor Calculus. Schaum's Outlines. McGraw Hill. pp. 18–19, 31–32. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-07-033484-6</bdi>.</cite></li>

<li><cite id="CITEREFM._R._SpiegelS._LipcshutzD._Spellman2009" class="citation book cs1">M. R. Spiegel; S. Lipcshutz; D. Spellman (2009). Vector analysis. Schaum's Outlines (2nd ed.). McGraw Hill. p. 227. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-07-161545-7</bdi>.</cite></li>

<li><cite id="CITEREFJ.R._Tyldesley1975" class="citation book cs1">J.R. Tyldesley (1975). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=PODXAAAAMAAJ">An introduction to tensor analysis for engineers and applied scientists</a>. Longman. pp. 5–13. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-582-44355-5</bdi>.</cite></li></ul>

<div class="mw-heading mw-heading3"><h3 id="Further_reading_and_applications">Further reading and applications</h3></div>

<ul><li><cite id="CITEREFS._LipcshutzM._Lipson2009" class="citation book cs1">S. Lipcshutz; M. Lipson (2009). Linear Algebra. Schaum's Outlines (4th ed.). McGraw Hill. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-07-154352-1</bdi>.</cite></li>

<li><cite id="CITEREFPei_Chi_Chou1992" class="citation book cs1">Pei Chi Chou (1992). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=9-pJ7Kg5XmAC&q=cartesian+tensor">Elasticity: Tensor, Dyadic, and Engineering Approaches</a>. Courier Dover Publications. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>048-666-958-0</bdi>.</cite></li>

<li><cite id="CITEREFT._W._Körner2012" class="citation book cs1">T. W. Körner (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=RO_TMc7ETPEC&q=cartesian+tensor">Vectors, Pure and Applied: A General Introduction to Linear Algebra</a>. Cambridge University Press. p. 216. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-11070-3356-6</bdi>.</cite></li>

<li><cite id="CITEREFR._Torretti1996" class="citation book cs1">R. Torretti (1996). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=vpW_sBxwr88C&q=cartesian+tensor&pg=PA103">Relativity and Geometry</a>. Courier Dover Publications. p. 103. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-4866-90466</bdi>.</cite></li>

<li><cite id="CITEREFJ._J._L._SyngeA._Schild1978" class="citation book cs1">J. J. L. Synge; A. Schild (1978). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=8vlGhlxqZjsC&q=cartesian+tensor&pg=PA127">Tensor Calculus</a>. Courier Dover Publications. p. 128. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-4861-4139-X</bdi>.</cite></li>

<li><cite id="CITEREFC._A._BalafoutisR._V._Patel1991" class="citation book cs1">C. A. Balafoutis; R. V. Patel (1991). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7BcpyUjmLpUC&q=cartesian+tensor">Dynamic Analysis of Robot Manipulators: A Cartesian Tensor Approach</a>. The Kluwer International Series in Engineering and Computer Science: Robotics: vision, manipulation and sensors. Vol. 131. Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0792-391-454</bdi>.</cite></li>

<li><cite id="CITEREFS._G._Tzafestas1992" class="citation book cs1">S. G. Tzafestas (1992). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=iG5W9IQhjd4C&q=cartesian+tensor&pg=PA45">Robotic systems: advanced techniques and applications</a>. Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-792-317-491</bdi>.</cite></li>

<li><cite id="CITEREFT._DassS._K._Sharma1998" class="citation book cs1">T. Dass; S. K. Sharma (1998). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=AQCsAxpZ7ToC&q=cartesian+tensor&pg=PA144">Mathematical Methods In Classical And Quantum Physics</a>. Universities Press. p. 144. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>817-371-0899</bdi>.</cite></li>

<li><cite id="CITEREFG._F._J._Temple2004" class="citation book cs1">G. F. J. Temple (2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=56WqzKbTMtMC&q=Cartesian+Tensors:+An+Introduction+temple">Cartesian Tensors: An Introduction</a>. Dover Books on Mathematics Series. Dover. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-4864-3908-9</bdi>.</cite></li>

<li><cite id="CITEREFH._Jeffreys1961" class="citation book cs1"><a href="Harold_Jeffreys" title="Harold Jeffreys">H. Jeffreys</a> (1961). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=oOYIAQAAIAAJ&q=cartesian+tensors">Cartesian Tensors</a>. Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780521054232</bdi>.</cite> <code class="cs1-code">{{cite book}}</code>: ISBN / Date incompatibility (help)</li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20120522142125/http://www.mso.anu.edu.au/~geoff/HEA/A1_Tensors.pdf">Cartesian Tensors</a></li>

<li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20120907020242/http://www.ce.udel.edu/faculty/kaliakin/appendix_tensors.pdf">V. N. Kaliakin, Brief Review of Tensors, University of Delaware</a></li>

<li><a rel="nofollow" class="external text" href="http://www.damtp.cam.ac.uk/user/reh10/lectures/nst-mmii-chapter3.pdf">R. E. Hunt, Cartesian Tensors, University of Cambridge</a></li></ul>

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</style><div id="Tensors176" style="font-size:114%;margin:0 4em"><a href="Tensor" title="Tensor">Tensors</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div><a href="Glossary_of_tensor_theory" title="Glossary of tensor theory">Glossary of tensor theory</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Scope</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Mathematics" title="Mathematics">Mathematics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Coordinate_system" title="Coordinate system">Coordinate system</a></li>

<li><a href="Differential_geometry" title="Differential geometry">Differential geometry</a></li>

<li><a href="Dyadics" title="Dyadics">Dyadic algebra</a></li>

<li><a href="Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a></li>

<li><a href="Exterior_calculus" class="mw-redirect" title="Exterior calculus">Exterior calculus</a></li>

<li><a href="Multilinear_algebra" title="Multilinear algebra">Multilinear algebra</a></li>

<li><a href="Tensor_algebra" title="Tensor algebra">Tensor algebra</a></li>

<li><a href="Tensor_calculus" class="mw-redirect" title="Tensor calculus">Tensor calculus</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><div class="hlist"><ul><li><a href="Physics" title="Physics">Physics</a></li><li><a href="Engineering" title="Engineering">Engineering</a></li></ul></div></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Computer_vision" title="Computer vision">Computer vision</a></li>

<li><a href="Continuum_mechanics" title="Continuum mechanics">Continuum mechanics</a></li>

<li><a href="Electromagnetism" title="Electromagnetism">Electromagnetism</a></li>

<li><a href="General_relativity" title="General relativity">General relativity</a></li>

<li><a href="Transport_phenomena" title="Transport phenomena">Transport phenomena</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Notation</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Abstract_index_notation" title="Abstract index notation">Abstract index notation</a></li>

<li><a href="Einstein_notation" title="Einstein notation">Einstein notation</a></li>

<li><a href="Index_notation" title="Index notation">Index notation</a></li>

<li><a href="Multi-index_notation" title="Multi-index notation">Multi-index notation</a></li>

<li><a href="Penrose_graphical_notation" title="Penrose graphical notation">Penrose graphical notation</a></li>

<li><a href="Ricci_calculus" title="Ricci calculus">Ricci calculus</a></li>

<li><a href="Tetrad_(index_notation)" class="mw-redirect" title="Tetrad (index notation)">Tetrad (index notation)</a></li>

<li><a href="Van_der_Waerden_notation" title="Van der Waerden notation">Van der Waerden notation</a></li>

<li><a href="Voigt_notation" title="Voigt notation">Voigt notation</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Tensor definitions</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Tensor_(intrinsic_definition)" title="Tensor (intrinsic definition)">Tensor (intrinsic definition)</a></li>

<li><a href="Tensor_field" title="Tensor field">Tensor field</a></li>

<li><a href="Tensor_density" title="Tensor density">Tensor density</a></li>

<li><a href="Tensors_in_curvilinear_coordinates" title="Tensors in curvilinear coordinates">Tensors in curvilinear coordinates</a></li>

<li><a href="Mixed_tensor" title="Mixed tensor">Mixed tensor</a></li>

<li><a href="Antisymmetric_tensor" title="Antisymmetric tensor">Antisymmetric tensor</a></li>

<li><a href="Symmetric_tensor" title="Symmetric tensor">Symmetric tensor</a></li>

<li><a href="Tensor_operator" title="Tensor operator">Tensor operator</a></li>

<li><a href="Tensor_bundle" title="Tensor bundle">Tensor bundle</a></li>

<li><a href="Two-point_tensor" title="Two-point tensor">Two-point tensor</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Operation_(mathematics)" title="Operation (mathematics)">Operations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Covariant_derivative" title="Covariant derivative">Covariant derivative</a></li>

<li><a href="Exterior_covariant_derivative" title="Exterior covariant derivative">Exterior covariant derivative</a></li>

<li><a href="Exterior_derivative" title="Exterior derivative">Exterior derivative</a></li>

<li><a href="Exterior_product" class="mw-redirect" title="Exterior product">Exterior product</a></li>

<li><a href="Hodge_star_operator" title="Hodge star operator">Hodge star operator</a></li>

<li><a href="Lie_derivative" title="Lie derivative">Lie derivative</a></li>

<li><a href="Raising_and_lowering_indices" class="mw-redirect" title="Raising and lowering indices">Raising and lowering indices</a></li>

<li><a href="Symmetrization" title="Symmetrization">Symmetrization</a></li>

<li><a href="Tensor_contraction" title="Tensor contraction">Tensor contraction</a></li>

<li><a href="Tensor_product" title="Tensor product">Tensor product</a></li>

<li><a href="Transpose" title="Transpose">Transpose</a> (2nd-order tensors)</li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related abstractions</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Affine_connection" title="Affine connection">Affine connection</a></li>

<li><a href="Basis_(linear_algebra)" title="Basis (linear algebra)">Basis</a></li>

<li><a href="Cartan_formalism_(physics)" class="mw-redirect" title="Cartan formalism (physics)">Cartan formalism (physics)</a></li>

<li><a href="Connection_form" title="Connection form">Connection form</a></li>

<li><a href="Covariance_and_contravariance_of_vectors" title="Covariance and contravariance of vectors">Covariance and contravariance of vectors</a></li>

<li><a href="Differential_form" title="Differential form">Differential form</a></li>

<li><a href="Dimension" title="Dimension">Dimension</a></li>

<li><a href="Exterior_form" class="mw-redirect" title="Exterior form">Exterior form</a></li>

<li><a href="Fiber_bundle" title="Fiber bundle">Fiber bundle</a></li>

<li><a href="Geodesic" title="Geodesic">Geodesic</a></li>

<li><a href="Levi-Civita_connection" title="Levi-Civita connection">Levi-Civita connection</a></li>

<li><a href="Linear_map" title="Linear map">Linear map</a></li>

<li><a href="Manifold" title="Manifold">Manifold</a></li>

<li><a href="Matrix_(mathematics)" title="Matrix (mathematics)">Matrix</a></li>

<li><a href="Multivector" title="Multivector">Multivector</a></li>

<li><a href="Pseudotensor" title="Pseudotensor">Pseudotensor</a></li>

<li><a href="Spinor" title="Spinor">Spinor</a></li>

<li><a href="Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">Vector</a></li>

<li><a href="Vector_space" title="Vector space">Vector space</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Notable tensors</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Mathematics</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Kronecker_delta" title="Kronecker delta">Kronecker delta</a></li>

<li><a href="Levi-Civita_symbol" title="Levi-Civita symbol">Levi-Civita symbol</a></li>

<li><a href="Metric_tensor" title="Metric tensor">Metric tensor</a></li>

<li><a href="Nonmetricity_tensor" title="Nonmetricity tensor">Nonmetricity tensor</a></li>

<li><a href="Ricci_curvature" title="Ricci curvature">Ricci curvature</a></li>

<li><a href="Riemann_curvature_tensor" title="Riemann curvature tensor">Riemann curvature tensor</a></li>

<li><a href="Torsion_tensor" title="Torsion tensor">Torsion tensor</a></li>

<li><a href="Weyl_tensor" title="Weyl tensor">Weyl tensor</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Physics</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Moment_of_inertia#Inertia_tensor" title="Moment of inertia">Moment of inertia</a></li>

<li><a href="Angular_momentum#Angular_momentum_in_relativistic_mechanics" title="Angular momentum">Angular momentum tensor</a></li>

<li><a href="Spin_tensor" title="Spin tensor">Spin tensor</a></li>

<li><a href="Cauchy_stress_tensor" title="Cauchy stress tensor">Cauchy stress tensor</a></li>

<li><a href="Stress%E2%80%93energy_tensor" title="Stress–energy tensor">stress–energy tensor</a></li>

<li><a href="Einstein_tensor" title="Einstein tensor">Einstein tensor</a></li>

<li><a href="Electromagnetic_tensor" title="Electromagnetic tensor">EM tensor</a></li>

<li><a href="Gluon_field_strength_tensor" title="Gluon field strength tensor">Gluon field strength tensor</a></li>

<li><a href="Metric_tensor_(general_relativity)" title="Metric tensor (general relativity)">Metric tensor (GR)</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Mathematician" title="Mathematician">Mathematicians</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="%C3%89lie_Cartan" title="Élie Cartan">Élie Cartan</a></li>

<li><a href="Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a></li>

<li><a href="Elwin_Bruno_Christoffel" title="Elwin Bruno Christoffel">Elwin Bruno Christoffel</a></li>

<li><a href="Albert_Einstein" title="Albert Einstein">Albert Einstein</a></li>

<li><a href="Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li>

<li><a href="Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a></li>

<li><a href="Hermann_Grassmann" title="Hermann Grassmann">Hermann Grassmann</a></li>

<li><a href="Tullio_Levi-Civita" title="Tullio Levi-Civita">Tullio Levi-Civita</a></li>

<li><a href="Gregorio_Ricci-Curbastro" title="Gregorio Ricci-Curbastro">Gregorio Ricci-Curbastro</a></li>

<li><a href="Bernhard_Riemann" title="Bernhard Riemann">Bernhard Riemann</a></li>

<li><a href="Jan_Arnoldus_Schouten" title="Jan Arnoldus Schouten">Jan Arnoldus Schouten</a></li>

<li><a href="Woldemar_Voigt" title="Woldemar Voigt">Woldemar Voigt</a></li>

<li><a href="Hermann_Weyl" title="Hermann Weyl">Hermann Weyl</a></li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

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