162 Covariant Rank
An integer counting how many lower indices a tensor carries that is used to track how those components change under a change of basis.
definition [d] (Covariant Rank = Covariant Degree) The integer \(s\) for a type-\((r,s)\) tensor: the number of vector arguments the multilinear map consumes,
- \(T^{r}_{\ s}(\omega^{1},\ldots,\omega^{r},\, \underbrace{v_{1},\ldots,v_{s}}_{s})\) .
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.
- \(s\) is the second slot in type-\((q,r)\) notation.
definition [d] (Covariant Rank = Covariant Degree) The number of subscript lower indices on the components of a tensor,
- \(T^{\mu_1\ldots\mu_r}_{\ \nu_1\ldots\nu_s}\) has covariant rank \(s\) ,
transforming covariantly 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.
162.1 Elementary Example
162.1.1 Simple
Covariant rank \(s\) counts lower indices. A covector is type \((0,1)\), so \(s = 1\).
\[ \omega = \omega_{1}\, e^{1} + \omega_{2}\, e^{2} \]
\[ r = 0,\quad s = 1 \]
where
- \(s\) is the covariant rank.
- \(r\) is the contravariant rank.
- \(\omega_{1}, \omega_{2}\) are the lower-index components of \(\omega\).
162.1.2 General
A type-\((0,2)\) tensor on \(\mathbb{R}^{3}\) has covariant rank \(s = 2\) and a \(3 \times 3\) matrix of components \(T_{ij}\).
\[ T : \mathbb{R}^{3} \times \mathbb{R}^{3} \rightarrow \mathbb{R} \]
\[ (T_{ij}) = \begin{pmatrix} T_{11} & T_{12} & T_{13} \\ T_{21} & T_{22} & T_{23} \\ T_{31} & T_{32} & T_{33} \end{pmatrix} \]
\[ r = 0,\quad s = 2 \]
where
- \(T_{ij}\) are the two lower-index components of \(T\).
162.2 References
- Hassani, S. Mathematical Physics, 2nd ed. Springer. — covariant degree \(s\) of a type-\((r,s)\) tensor.
- Arfken, G. B., Weber, H. J., & Harris, F. E. Mathematical Methods for Physicists, 7th ed. Elsevier / Academic Press, 2013. — covariant rank equals the number of lower indices.
- Frankel, T. The Geometry of Physics, 3rd ed. Cambridge University Press. — covariant arguments and indices.
- Alternating Function
- Alternating k-linear function
- Boundary Operation
- Bundle
- Cartesian Product
- Constant Metric Tensor
- Contravariant Rank
- Covariant Rank
- Covariant, Contravariant, and Coordinate Systems
- Differential Forms
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- Edge of a Domain
- Exterior Algebra and Exterior Derivatives
- Forms
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- Multilinear Function
- Nondegenerate
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- Stoke’s Theorem and the Fundamental Theorem of Calculus
- Symmetric Array
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- Type-(0,2) Tensor Field
- Type-(q,r) Tensor