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Mathematics for Physics

161 Contravariant Rank

An integer counting how many upper indices a tensor carries that is used to track how those components change under a change of basis.

definition [d] (Contravariant Rank = Contravariant Degree) The integer \(r\) for a type-\((r,s)\) tensor: the number of dual-vector arguments the multilinear map consumes,

  • \(T^{r}_{\ s}(\underbrace{\omega^{1},\ldots,\omega^{r}}_{r},\, v_{1},\ldots,v_{s})\) .

where

  • \(T^{r}_{\ s}\) is a tensor of type \((r,s)\).
  • \(r\) is the contravariant rank.
  • \(s\) is the covariant rank.
  • \(\omega^{1},\ldots,\omega^{r}\) are covectors in \(V^{*}\).
  • \(v_{1},\ldots,v_{s}\) are vectors in \(V\).
  • \(V\) is a vector space.
  • \(V^{*}\) is the dual of \(V\).

Note:

  • \(r\) is also called the contravariant degree.
  • \(s\) is also called the covariant degree.
  • \(r\) is the first slot in type-\((q,r)\) notation when \(q = r\).

definition [d] (Contravariant Rank = Contravariant Degree) The number of superscript upper indices on the components of a tensor,

  • \(T^{\mu_1\ldots\mu_r}_{\ \nu_1\ldots\nu_s}\) has contravariant rank \(r\) ,

transforming contravariantly under a change of coordinates.

where

  • \(T^{\mu_1\ldots\mu_r}_{\ \nu_1\ldots\nu_s}\) are the components of the tensor.
  • \(\mu_1,\ldots,\mu_r\) are the upper, contravariant indices.
  • \(\nu_1,\ldots,\nu_s\) are the lower, covariant indices.
  • \(r\) is the contravariant rank.
  • \(s\) is the covariant rank.

Note:

  • the total rank is \(r + s\).
  • the total rank is also called the order.

161.1 Elementary Example

161.1.1 Simple

Contravariant rank \(r\) counts upper indices. A vector is type \((1,0)\), so \(r = 1\).

\[ v = v^{1} e_{1} + v^{2} e_{2} \]

\[ r = 1,\quad s = 0 \]

where

  • \(r\) is the contravariant rank.
  • \(s\) is the covariant rank.
  • \(v^{1}, v^{2}\) are the upper-index components of \(v\).

161.1.2 General

A type-\((2,0)\) tensor on \(\mathbb{R}^{3}\) has contravariant rank \(r = 2\) and nine components \(T^{ij}\).

\[ T = \sum_{i,j=1}^{3} T^{ij}\, e_{i} \otimes e_{j} \]

\[ (T^{ij}) = \begin{pmatrix} T^{11} & T^{12} & T^{13} \\ T^{21} & T^{22} & T^{23} \\ T^{31} & T^{32} & T^{33} \end{pmatrix} \]

\[ r = 2,\quad s = 0 \]

where

  • \(T^{ij}\) are the two upper-index components of \(T\).
  • \(e_{i} \otimes e_{j}\) is the basis tensor with those upper indices.

161.2 References

  1. Hassani, S. Mathematical Physics, 2nd ed. Springer. — contravariant degree \(r\) of a type-\((r,s)\) tensor.
  2. Arfken, G. B., Weber, H. J., & Harris, F. E. Mathematical Methods for Physicists, 7th ed. Elsevier / Academic Press, 2013. — contravariant rank equals the number of upper indices.
  3. Frankel, T. The Geometry of Physics, 3rd ed. Cambridge University Press. — contravariant arguments and indices.