Contraction of the tensor $T(u,v,\phi, \theta) = \phi (u) ^{.} \theta (v)$ $\;$?

Question

How to write all the possible contractions of a tensor $(2,2)$ (type of the tensor) which is defined as

$T(u,v,\phi, \theta) = \phi (u) ^{.} \theta (v)$ $\;$?

This expression is from my textbook and I dont understand it at all, every tensor I have ever encountered looked very differently.

Answer 1

$u$ and $v$ are members of some vector space $V$ over a field $F$, and $\phi$ and $\theta$ are members of the corresponding dual vector space $V^*$. So $T$ is a linear map

$T:V \times V \times V^* \times V^* \rightarrow F$

such that

$T(u,v,\phi,\theta) = \phi(u)\theta(v)$

Since $T$ is a $(2,2)$ tensor, we can write it in component form as $T^{ab}_{cd}$. Relative to a given basis $\{e_i\}$ of $V$ and the corresponding dual basis $\{e^j\}$ of $V^*$, the components of $T$ are

$T^{ab}_{cd}=T(e_c,e_d,e^a,e^b)=e^a(e_c)e^b(e_d)=\delta^a_c\delta^b_d$

We can contract either of the upper indices with either of the lower indices to get four possible contractions

$T^{jb}_{jd}, \space T^{jb}_{cj}, \space T^{aj}_{jd}, \space T^{aj}_{cj}$

or we can contract both upper indices with both lower indices in two different ways to get:

$T^{jk}_{jk}, \space T^{jk}_{kj}$