Tensor field is important to the area of physics study. This note will make some brief introduction to it.

Tensor is a physical value induced to describe the property of anisotropy media. In normal condition it is a 2 order tensor by default, mathematically it is equal to a 2 order matrix. What’s more the vector could also be treated as a one order tensor. Comparing with the high order matrix, we can further define a high order tensor. And all the mathematical properties of matrix applies also to the tensor.

  • Definition of 2 order tensor

M⃗⃗=(M11M12M13M21M22M23M31M32M33)=∑ijMijei⃗ej⃗\vec{\vec{M}}=\left ( \begin{array}{l l l} M_{11} & M_{12} & M_{13}\\ M_{21} & M_{22} & M_{23}\\ M_{31} & M_{32} & M_{33}\\ \end{array} \right ) =\sum_{ij}M_{ij}\vec{e_i}\vec{e_j}

But tensor has some difference from matrix. For a tensor, every element is actually linked to a dyad that represents its spacial direction. Comparing with 2 order tensor, 3 order tensor can be defined as M=∑ijkMijkei⃗ej⃗ek⃗M=\sum_{ijk}M_{ijk}\vec{e_i}\vec{e_j}\vec{e_k}, tensors of other order can also be defined in this way. Such as: 0 th tensor is a scaler, 1 st order tensor is a vector, … , … n th order tensor.

  • Dyad
    Dyad is a class of special 2 nd order tensor, it is the result of two vector after dyadic operation.

A⃗B⃗=(AxAyAz)(Bx,By,Bz)=(AxBxAxByAxBzAyBxAyByAyBzAzBxAzByAzBz)=∑ijAiBjei⃗ej⃗\begin{array}{l} \vec{A}\vec{B} & = \left ( \begin{array}{l} A_x\\ A_y\\ A_z \end{array} \right ) (B_x, B_y, B_z)\\ & = \left ( \begin{array}{l l l} 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 \end{array} \right ) =\sum_{ij}A_i B_j\vec{e_i}\vec{e_j} \end{array}

For the case of directional dyad ei⃗ej⃗\vec{e_i}\vec{e_j}, it is equal to a tensor 2 nd order tensor with all zero element but one non-zeor element.
eg: M=ex⃗ey⃗M=\vec{e_x}\vec{e_y} only has one non-zero elementM12=1M_{12}=1。

ex⃗ey⃗=(010000000)\vec{e_x}\vec{e_y} =\left ( \begin{array}{l l l} 0 & 1 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{array} \right )

Pay attention, dyad is mathematically a matrix. You can not exchange the order of the two components. And the normal tensors do not satisfy the rule of exchange for multiplication.

Calculation rules of tensor

Combination (nearby rule)

A⃗⋅B⃗C⃗=(A⃗⋅B⃗)C⃗B⃗C⃗⋅A⃗=B⃗(C⃗⋅A⃗)\begin{array}{l} \vec{A}\cdot\vec{B}\vec{C}=(\vec{A}\cdot\vec{B})\vec{C}\\ \vec{B}\vec{C}\cdot\vec{A}=\vec{B}(\vec{C}\cdot\vec{A}) \end{array}

This calculation shows that A⃗\vec{A} dot a tensor can get a vector, whose direction may not in the direction of A⃗\vec{A}.

Differential operation to produce a tensor

∇A⃗=(ex⃗∂∂x+ey⃗∂∂y+ez⃗∂∂z)(Axex⃗+Ayey⃗+Azez⃗)=(∂∂x∂∂y∂∂z)(Ax,Ay,Az)=(∂Ax∂x∂Ay∂x∂Az∂x∂Ax∂y∂Ay∂y∂Az∂y∂Ax∂z∂Ay∂z∂Az∂z)=∑ij∂Ai∂xjej⃗ei⃗\begin{array}{l} \nabla\vec{A} & =(\vec{e_x}\frac{\partial}{\partial x}+\vec{e_y}\frac{\partial}{\partial y}+\vec{e_z}\frac{\partial}{\partial z})(A_x\vec{e_x}+A_y\vec{e_y}+A_z\vec{e_z})\\ & =\left ( \begin{array}{l l l} \frac{\partial}{\partial x}\\ \frac{\partial}{\partial y}\\ \frac{\partial}{\partial z} \end{array} \right ) (A_x, A_y, A_z) =\left ( \begin{array}{l l l} \frac{\partial A_x}{\partial x} & \frac{\partial A_y}{\partial x} & \frac{\partial A_z}{\partial x}\\ \frac{\partial A_x}{\partial y} & \frac{\partial A_y}{\partial y} & \frac{\partial A_z}{\partial y}\\ \frac{\partial A_x}{\partial z} & \frac{\partial A_y}{\partial z} & \frac{\partial A_z}{\partial z} \end{array} \right )\\ & =\sum_{ij}\frac{\partial A_i}{\partial x_j}\vec{e_j}\vec{e_i} \end{array}

Besides the ∇\nabla operator, similar to dyad, we can define high order derivative operators like ∇∇\nabla\nabla. ∇∇A⃗\nabla\nabla\vec{A} is a 3 th oder tensor. Every time after a ∇\nabla operator working on a tensor, its order grows by one.

Contraction of Tensor

After a contraction between two tensors, two 2 nd tensors will produce one 2 order tensor:
A⃗B⃗⋅C⃗D⃗=A⃗(B⃗⋅C⃗)D⃗=(B⃗⋅C⃗)A⃗D⃗\vec{A}\vec{B}\cdot\vec{C}\vec{D}=\vec{A}(\vec{B}\cdot\vec{C})\vec{D}=(\vec{B}\cdot\vec{C})\vec{A}\vec{D}

Two 2 nd tensors after 2 contraction operation will produce a 0 th order tensor (scaler):
A⃗B⃗:C⃗D⃗=(B⃗⋅C⃗)(A⃗⋅D⃗)\vec{A}\vec{B}:\vec{C}\vec{D}=(\vec{B}\cdot\vec{C})(\vec{A}\cdot\vec{D})
Similarly, we can also define higher order contraction operations.

Operations of unit tensor

A⃗⋅I⃗⃗=A⃗\vec{A}\cdot\vec{\vec{I}}=\vec{A}

M⃗⃗⋅I⃗⃗=M⃗⃗\vec{\vec{M}}\cdot\vec{\vec{I}}=\vec{\vec{M}}

M⃗⃗:I⃗⃗=trace(M⃗⃗)\vec{\vec{M}}:\vec{\vec{I}}=trace(\vec{\vec{M}})

Tyler expansion of multi-elements function

For multi-elements functiion, we can make use of the dyad tensor to simplify the derivative operator. Thus we can avoid the complex writing in different coordinate systems.

f(r0⃗+Δr⃗)=∑n(Δr⃗⋅∇)(n)n!f(n)(r0⃗)=f(r0⃗)+Δr⃗⋅∇f(r0⃗)+12!Δr⃗Δr⃗:∇∇f(r0⃗)+...\begin{array}{l l} f(\vec{r_0}+\Delta\vec{r}) & =\sum_n \frac{(\Delta\vec{r}\cdot\nabla)^{(n)}}{n!}f^{(n)}(\vec{r_0})\\ & =f(\vec{r_0})+\Delta\vec{r}\cdot\nabla f(\vec{r_0})+\frac{1}{2!}\Delta\vec{r}\Delta\vec{r}:\nabla\nabla f(\vec{r_0})+ ... \end{array}

This clever way makes use of the rules of dyad and contraction, greatly simplified effect to extract the operations to each element.