Rank

Relevant parts to questions...

• Count free indices to find the rank - dummy (repeated) indices don’t count.
• A rank-$r$ tensor in 3D has $3^r$ components and needs exactly $r$ factors of $L$ in its Tensor Transformation Rule.
• Contraction drops rank by 2, Outer Product adds ranks together, Tensor Addition preserves rank.

The rank (or order) of a tensor is the number of free indices. Dummy (summed/repeated) indices don’t contribute. A rank-$r$ tensor in 3D has $3^r$ components and transforms with exactly $r$ factors of the rotation matrix in its Tensor Transformation Rule.

Rank	Object	Examples
0	Scalar	Temperature, $f$, $\mathbf{a}\cdot \mathbf{b}=a_i{b}_i$, trace $A_{ii}$
1	Vector	Position $x_i$, gradient $\partial f/\partial x_i$
2	Matrix/tensor	Metric Tensor $g_{ij}$, $[\nabla \mathbf{u}{]}_{ij}$, $A_{ik}{B}_{kj}$
3	Cube	Alternating Tensor ${ϵ}_{ijk}$, Christoffel-like objects
4	Hyper-array	Riemann-Christoffel Tensor $R_{ijk}^r$

Rank arithmetic (key for the exam):

• Outer Product: rank-$p$ $\otimes$ rank-$q$ = rank-$(p+q)$. Example: $A_{ij}{B}_{k\ell }$ is rank 4.
• Contraction: pairing two indices drops rank by 2. Example: $T_{ijij}$ is rank 0 (scalar) from rank 4.
• Inner product: outer product + contraction. Example: $A_{ij}{B}^i$ is rank 1.

Example

$T^i{}_p{}^j{R}_{ij}{}^q$ - $T$ is rank 3, $R$ is rank 3, outer product gives rank 6, then $i,i$ and $j,j$ contractions give rank $6-4=2$. Free indices left: $p$ and $q$. Rank 2.