In geometry and linear algebra, a Cartesian tensor uses an orthonormal basis to represent a tensor in a Euclidean space in the form of components. Converting a tensor's components from one such basis to another is done through an orthogonal transformation. The most familiar coordinate systems are the two-dimensional and three-dimensional Cartesian coordinate systems. Cartesian tensors may be used with any Euclidean space, or more technically, any finite-dimensional vector space over the field of real numbers that has an inner product. Use of Cartesian tensors occurs in physics and engineering, such as with the Cauchy stress tensor and the moment of inertia tensor in rigid body dynamics. Sometimes general curvilinear coordinates are convenient, as in high-deformation continuum mechanics, or even necessary, as in general relativity. While orthonormal bases may be found for some such coordinate systems (e.g. tangent to spherical coordinates), Cartesian tensors may provide considerable simplification for applications in which rotations of rectilinear coordinate axes suffice. The transformation is a passive transformation, since the coordinates are changed and not the physical system.
Cartesian basis and related terminology
Vectors in three dimensions In 3D Euclidean space, R 3 {\displaystyle \mathbb {R} ^{3}} , the standard basis is ex, ey, ez. Each basis vector points along the x-, y-, and z-axes, and the vectors are all unit vectors (or normalized), so the basis is orthonormal. Throughout, when referring to Cartesian coordinates in three dimensions, a right-handed system is assumed and this is much more common than a left-handed system in practice, see orientation (vector space) for details. For Cartesian tensors of order 1, a Cartesian vector a can be written algebraically as a linear combination of the basis vectors ex, ey, ez:
a = a x e x + a y e y + a z e z {\displaystyle \mathbf {a} =a_{\text{x}}\mathbf {e} _{\text{x}}+a_{\text{y}}\mathbf {e} _{\text{y}}+a_{\text{z}}\mathbf {e} _{\text{z}}}
where the coordinates of the vector with respect to the Cartesian basis are denoted ax, ay, az. It is common and helpful to display the basis vectors as column vectors
e x = ( 1 0 0 ) , e y = ( 0 1 0 ) , e z = ( 0 0 1 ) {\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}}}
when we have a coordinate vector in a column vector representation:
a = ( a x a y a z ) {\displaystyle \mathbf {a} ={\begin{pmatrix}a_{\text{x}}\\a_{\text{y}}\\a_{\text{z}}\end{pmatrix}}}
A row vector representation is also legitimate, although in the context of general curvilinear coordinate systems the row and column vector representations are used separately for specific reasons – see Einstein notation and covariance and contravariance of vectors for why. The term "component" of a vector is ambiguous: it could refer to:
a specific coordinate of the vector such as az (a scalar), and similarly for x and y, or 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. A more general notation is tensor index notation, 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:
a = a 1 e 1 + a 2 e 2 + a 3 e 3 = ∑ i = 1 3 a i e i {\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}}
It is standard to use the Einstein notation—the summation sign for summation over an index that is present exactly twice within a term may be suppressed for notational conciseness:
a = ∑ i = 1 3 a i e i ≡ a i e i {\displaystyle \mathbf {a} =\sum _{i=1}^{3}a_{i}\mathbf {e} _{i}\equiv a_{i}\mathbf {e} _{i}}
An advantage of the index notation over coordinate-specific notations is the independence of the dimension of the underlying vector space, i.e. the same expression on the right hand side takes the same form in higher dimensions (see below). Previously, the Cartesian labels x, y, z were just labels and not indices. (It is informal to say "i = x, y, z").
Second-order tensors in three dimensions A dyadic tensor T is an order-2 tensor formed by the tensor product ⊗ 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):
T = ( a x e x + a y e y + a z e z ) ⊗ ( b x e x + b y e y + b z e z )
= a x b x e x ⊗ e x + a x b y e x ⊗ e y + a x b z e x ⊗ e z
+
a y b x e y ⊗ e x + a y b y e y ⊗ e y + a y b z e y ⊗ e z
+
a z b x e z ⊗ e x + a z b y e z ⊗ e y + a z b z e z ⊗ e z {\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}}}
Representing each basis tensor as a matrix:
e x ⊗ e x ≡ e xx = ( 1 0 0 0 0 0 0 0 0 ) , e x ⊗ e y ≡ e xy = ( 0 1 0 0 0 0 0 0 0 ) , e z ⊗ e z ≡ e zz = ( 0 0 0 0 0 0 0 0 1 ) {\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}}}
then T can be represented more systematically as a matrix:
T = ( a x b x a x b y a x b z a y b x a y b y a y b z a z b x a z b y a z b z ) {\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}}}
See matrix multiplication 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:
T = T xx e xx + T xy e xy + T xz e xz
+
T yx e yx + T yy e yy + T yz e yz
+
T zx e zx + T zy e zy + T zz e zz {\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}}}
while in terms of tensor indices:
T = T i j e i j ≡ ∑ i j T i j e i ⊗ e j , {\displaystyle \mathbf {T} =T_{ij}\mathbf {e} _{ij}\equiv \sum _{ij}T_{ij}\mathbf {e} _{i}\otimes \mathbf {e} _{j}\,,}
and in matrix form:
T = ( T xx T xy T xz T yx T yy T yz T zx T zy T zz ) {\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}}}
Second-order tensors occur naturally in physics and engineering when physical quantities have directional dependence in the system, often in a "stimulus-response" way. This can be mathematically seen through one aspect of tensors – they are multilinear functions. A second-order tensor T which takes in a vector u of some magnitude and direction will return a vector v; of a different magnitude and in a different direction to u, in general. The notation used for functions in mathematical analysis leads us to write v − T(u), while the same idea can be expressed in matrix and index notations (including the summation convention), respectively:
( v x v y v z ) = ( T xx T xy T xz T yx T yy T yz T zx T zy T zz ) ( u x u y u z ) , v i = T i j u j {\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}}}
By "linear", if u = ρr + σs for two scalars ρ and σ and vectors r and s, then in function and index notations:
v = T ( ρ r + σ s ) =