Problem Two
(3 points)
Consider the tensor $T ::= V \otimes V \otimes V \otimes V^*$, which can be expanded in a
basis as
$T = T^{ijk}_{m} e_i \otimes e_j \otimes e_k \otimes e^m$
where we are using Einstein's summation conventions.
(2.1)
(2.1) (a)
Assume that $V$ is a 2-dimensional vector space over real numbers. In
fact, the nonzero components are known to be
$T^{111}_1 = 7$, $T^{121}_1 = 3$, $T^{112}_2 = -5$, $T^{211}_2 = -1$,
(2.2)
How many different tensors of the type $V \otimes V$ can we obtain by con-
tracting indices of $T$? Compute the components of all such tensors!
(2.2) (b)
Assume that $V$ is now a 3-dimensional vector space over real num-
bers, and the only nonzero components are those in (2.2). Compute
the value of the scalar $a$ defined as
$a = T^{ijk}_{m} \eta_{ij} \delta^m_k$
(2.3)
for the object $\eta_{ij}$ for which $\eta_{11} = -1$, $\eta_{22} = \eta_{33} = 1$, and $\eta_{ij} = 0$ for
$i \neq j$. Here, $\delta$ is the Kronecker symbol.
(2.3) (c)
Let us stick to a 3-dimensional vector space over real numbers, but
increase the nonzero components of $T$ as adding the following to the
list in (2.2):
(2.4)
$T^{133}_1 = 17$, $T^{321}_1 = -13$, $T^{132}_2 = 1$, $T^{322}_2 = -2$,
Write the explicit expression for the covector $\omega = \epsilon_{ijk} T^{ijk}_m e^m$ where
we will take basis vectors as $e^1 = dx$, $e^2 = dy$, and $e^3 = dz$. Here, $\epsilon$ is
the Levi-Civita symbol
