Tensor
#Math
Number elements in tensor $\displaystyle =\mathrm{dimension^{rank}}$
- E.g. the 3D stress tensor is a rank $2$ tensor and has $3^2$ components. Likewise, vectors in 3D are rank $1$ tensors and have $3^1$ components
- Tensor rank is the number of subscripts needed to specify an element (need to check this)
- A tensor's dimension is the space the tensor lives in (need to check this)
$T_{ij}'=R_{ik}R_{jk}T_{kl}$
- Describes how a rank 2 tensor transforms under two rotations
- Symmetric:
- $T_{ij}=T_{ji}$
- Antisymmetric:
- $T_{ij}=-T_{ji}$
Tensor Definition
- Best overview video explanation
- Best contravariant/covariant explanation: https://www.youtube.com/watch?v=CliW7kSxxWU
- Tensor explanation 1: https://www.youtube.com/watch?v=7c8Agf9qtfI
- Tensor explanation 2: https://www.youtube.com/watch?v=ztUHlZftPlo
- Tensor explanation 3: https://www.youtube.com/watch?v=nNMY02udkHw
- A rank $n$ tensor in $m$ dimensional space has $n$ indicies (or coordinates), $m^n$ components, and transforms as described by a generalized version of the above transformation rule
- E.g. the 3D stress tensor is a rank $2$ tensor and has $3^2$ components. Likewise, vectors in 3D are rank $1$ tensors and have $3^1$ components
- Superscript represents column vectors:
$$T^i = (T^i){i = 1, 2, 3} =
\begin{bmatrix}
T^1\\ T^2\\ T^3
\end{bmatrix}$$
Subscript represents row vectors:
$$T{i} = (T_i)_{i = 1, 2, 3} = \begin{bmatrix}
T_1 & T_2 & T_3
\end{bmatrix}$$
| Contravariant Components | Covariant Components | |
|---|---|---|
| As basis vector length increases | Tensor components decrease | Tensor components increase |
| Script | Superscript | Subscript |
| Assuming: | $\vec v = \left(\frac{\partial x^1}{dt}, \frac{\partial x^2}{dt}, \ldots, \frac{\partial x^n}{dt}\right)$ | $\vec \nabla F = (u_1, u_2, \ldots, u_n), u_i = \frac{\partial F}{\partial x^i}$ |
| Transformation | $\bar v^i = v^r\frac{\partial \bar x^i}{\partial x^r}$ | $\bar u_i = u_r\frac{\partial x^r}{\partial\bar x^i}$, where $u$ is a covariant first rank tensor in the that transforms from $u_i$ in the $(x^i)$ coordinate system to $\bar u_i$ in the $(\bar x^i)$ coordinate system |
| Projection | Parallel Projection | Perpendicular Projection |
| Coordinates | $\vec a = a^1\hat e_1 + a^2 \hat e_2$ | $\vec a = a_1\hat e_1 + a_2 \hat e_2$ |
| Examples | Position, velocity, acceleration, etc. | Gradient of scalar function |