Section 3.1 Rank of a Tensor
The rank (order) of a tensor is the number of indices in the symbol representing a tensor. For example: \(A\) is a tensor of rank zero (scalar), \(B^{i}\) is a contravarient tensor of rank one (vector), \(C_{k}\) is a covarient tensor of rank one (vector), \(D_{ij}\) is a covarient tensor of rank two (dyadic), \(M^{p q r}\) is a contravarient tensor of rank three (triadic), and \(T_{kl.....}^{ij....}\) is a mixed tensor of rank four.
In an \(n\)- dimensional space, the number of components of a tensor of rank \(r\) is \(n^{r}\text{.}\) For example: A scalar has one component (magnitude only) and hence zero basis vector per component. A vector has two components (magnitude and one direction) in \(2D\) and hence has 1 basis vector per component. A vector has 3 components (magnitude and one direction) in 3D and hence has 1 basis vector per component. A tensor of rank 2 (dyad) has \(3^{2} = 9\) components (magnitude and two directions) in 3D hence has 2 basis vectors per component. A tensor of rank 3 (triad) has \(3^{3} = 27\) components (magnitude and three directions) in 3D hence has 3 basis vectors per component and so on.
