Tensor Field
Definition
In physics, a tensor field is a function that assigns a tensor value to each vector (point) in space, i.e. a function of the following form.
$$ f: \mathbb{R}^{n} \to \mathbb{R}^{m \times p} $$
Explanation
A function called a field indicates that the elements of its domain are vectors. Thus a tensor field is a function that maps vectors to tensors.
| 함수 | 대응관계 | 예시 |
|---|---|---|
| 스칼라 필드 | 벡터 $\mapsto$ 🔒(26/04/27)스칼라 | 점 $(x, y, z)$에서의 온도 $T = T(x,y,z)$ |
| 벡터 필드 | 벡터 $\mapsto$ 벡터 | 점 $(x, y, z)$에서 물체의 속도 $\mathbf{v} = \mathbf{v}(x, y, z) = \begin{bmatrix} v_{x}(x,y,z) \\ v_{y}(x,y,z) \\ v_{z}(x,y,z) \end{bmatrix}$ |
| 텐서 필드 | 벡터 $\mapsto$ 텐서 | 점 $(x, y, z)$에서 물체가 받는 응력 $\sigma = \sigma(x, y, z) = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\ \sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\ \sigma_{zx} & \sigma_{zy} & \sigma_{zz} \end{bmatrix}$ |
Examples include the following.
Maxwell field strength tensor: $$ \mathbf{T} = \begin{bmatrix} T_{xx} & T_{xy} & T_{xz} \\ T_{yx} & T_{yy} & T_{yz} \\ T_{zx} & T_{zy} & T_{zz} \end{bmatrix} $$ $$ T_{ij} = \epsilon_{0} \left( E_{i}E_{j}-\dfrac{1}{2}\delta_{ij}E^2 \right) + \dfrac{1}{\mu_{0}}\left(B_{i}B_{j}-\dfrac{1}{2}\delta_{ij}B^2 \right) $$
Cauchy stress tensor: $$ \sigma = \begin{bmatrix} \sigma_{11} & \sigma_{12} & \sigma_{13} \\ \sigma_{21} & \sigma_{22} & \sigma_{23} \\ \sigma_{31} & \sigma_{32} & \sigma_{33} \end{bmatrix} = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\ \sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\ \sigma_{zx} & \sigma_{zy} & \sigma_{zz} \end{bmatrix} $$
Moment of inertia tensor: $$ I = \begin{bmatrix} I_{11} & I_{12} & I_{13} \\ I_{21} & I_{22} & I_{23} \\ I_{31} & I_{32} & I_{33} \end{bmatrix} = \begin{bmatrix} I_{xx} & I_{xy} & I_{xz} \\ I_{yx} & I_{yy} & I_{yz} \\ I_{zx} & I_{zy} & I_{zz} \end{bmatrix} $$