A rank n tensor in m dimensional space has n indicies (or coordinates), mn 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 32 components. Likewise, vectors in 3D are rank 1 tensors and have 31 components
Superscript represents column vectors:
\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 |