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Mathematics for Physics

196 Type-(q,r) Tensor

A multilinear mapping that takes a fixed number of covectors and vectors to a scalar that is used to classify tensors by index type.

definition [d] (Type-\((q,r)\) Tensor = Type-\((r,s)\) Tensor = Rank \((r,s)\)) A multilinear map taking \(r\) covectors and \(s\) vectors to a scalar,

  • \(T^{r}_{\ s} : \underbrace{V^{*}\times\cdots\times V^{*}}_{r} \times \underbrace{V\times\cdots\times V}_{s} \rightarrow \mathbb{R}\) .

where

  • \(T^{r}_{\ s}\) is a tensor of type \((r,s)\).
  • \(V\) is a vector space.
  • \(V^{*}\) is the dual space of \(V\).
  • \(r\) is the contravariant degree.
  • \(s\) is the covariant degree.
  • \(\mathbb{R}\) is the set of real numbers.

Note:

  • type \((r,s)\) uses the same ordering as type \((q,r)\) with \(q = r\) and the second slot equal to \(s\).
  • type \((1,0)\) is a vector.
  • type \((0,1)\) is a covector.
  • type \((0,1)\) is also called a \(1\)-form.
  • the space of such tensors at a point \(P\) is denoted \(T^{r}_{\ s,P}(M)\).

definition [d] (Type-\((q,r)\) Tensor = Rank-\(n\) Tensor = Tensorial Set) A system of components with \(r\) upper and \(s\) lower indices that transform by the tensor law under coordinate changes,

  • \(T^{\mu_1\ldots\mu_r}_{\ \nu_1\ldots\nu_s}\) ,

with total rank \(n = r + s\).

where

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

Note:

  • \(n\) is also called the order.
  • upper indices transform contravariantly; lower indices covariantly.
  • also called valence \(\{r,s\}\) in some texts.

196.1 Elementary Example

196.1.1 Simple

A type-\((0,1)\) tensor is a covector: one lower index and no upper index.

\[ T : V \rightarrow \mathbb{R} \]

\[ V = \{ e_{1},\ e_{2},\ e_{3} \} \]

\[ T(e_{1}) = 1,\quad T(e_{2}) = 0,\quad T(e_{3}) = -1 \]

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

where

  • \(r\) is the contravariant degree.
  • \(s\) is the covariant degree.

196.1.2 General

A type-\((1,1)\) tensor on \(\mathbb{R}^{3}\) is a \(3 \times 3\) matrix acting as a linear map on components.

\[ T : V^{*} \times V \rightarrow \mathbb{R} \]

\[ (T^{i}_{\ j}) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{pmatrix} \]

\[ T^{i}_{\ j}\, v^{j} = w^{i} \]

where

  • \(T^{i}_{\ j}\) are the mixed components of a type-\((1,1)\) tensor.
  • \(r = 1\) and \(s = 1\).

196.2 References

  1. Hassani, S. Mathematical Physics, 2nd ed. Springer. — type-\((r,s)\) tensor as multilinear map \(V^{*r}\times V^{s}\to\mathbb{R}\).
  2. Arfken, G. B., Weber, H. J., & Harris, F. E. Mathematical Methods for Physicists, 7th ed. Elsevier / Academic Press, 2013. — rank-\(n\) tensors; component transformation laws.
  3. Frankel, T. The Geometry of Physics, 3rd ed. Cambridge University Press. — multilinear tensor algebra on manifolds.