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\newpage
\section{Tensors}

\subsection{What is a Tensor?}

Not any list of $n$ numbers constitutes a vector
in $\RR^n$. They come with certain
responsibilities. \myskip
We start with a point $\bf{x} \in \RR^n$. To
attach some coordinates to this, we first
introduce a basis $\{\bf{e}_i, i = 1, \dots, n\}$
such that
\[ \bf{e}_i \cdot \bf{e}_j = \delta_{ij} \]
And we write $\bf{x} = x_i \bf{e}_i$. We call $x_i
= (x_1, \dots, x_n)$ a ``vector''. It's a set of
labels to specify $\bf{x}$. \myskip
Alternatively, we could use
\[ \bf{e}_i' = R_{ij} \bf{e}_j \tag{$*$} \]
We insist that $\bf{e}_i' \cdot \bf{e}_j' =
\delta_{ij}$.
\[ \implies R_{ik}R_{jk} \bf{e}_k \cdot \bf{e}_l =
R_{ik}R_{jk} = \delta_{ij} \]
\[ \implies RR^\top = \mathbbm{1} \]
Such matrices are called \emph{orthogonal}. We
write $R \in O(n)$. We have
\[ \det RR^\top = (\det R)^2 = 1 \implies \det R =
\pm 1 \]
If $\det R = + 1$, then $R$ corresponds to a
rotation and we write $R \in SO(n)$ (special
orthogonal).  \\
If $\det R = -1$, it is a reflection + rotation.
Under a change of basis, $\bf{x}$ doesn't change.
We have
\[ \bf{x} = x_i \bf{e}_i = x_i' \bf{e}_i' = x_i'
R_{ij} \bf{e}_j \]
\[ \bf{x} \cdot \bf{e}_k = x_k = x_i' R_{ik} \]
inverting:
\[ \implies x_i' = R_{ij} x_j \]
\myskip
A \emph{tensor} $T$ is a generalisation of these
ideas to an object with more indices. When
measured with respect to the basis $\{\bf{e}_i\}$,
a \emph{tensor of rank $p$} (or \emph{$p$-tensor})
has indices
\[ T_{i_1 \cdots i_p} \]
Under a change of basis ($*$) we have the
\emph{tensor transformation rule}
\[ T_{i_1 \cdots i_p}' = R_{i_1j_1} \cdots
R_{i_pj_p} T_{j_1 \cdots j_p} \]

\begin{note*}
    0-tensor is a number \\
    1-tensor is a vector \\
    2-tensor is a matrix such that $T_{ij}' =
    R_{ik}R_{jl} T_{kl}$.
\end{note*}

\begin{example*}
    There is one special rank 2 tensor in $\RR^n$:
    \[ \delta_{ij} = \begin{cases}
        1 & i = j \\
        0 & \text{otherwise}
    \end{cases} \]
    This is the same in all bases since
    \[ \delta_{ij}' = R_{ik}R_{jl}\delta_{kl} =
    \delta_{ij} \]
    It's an example of an \emph{invariant} tensor.
\end{example*}

\subsubsection*{Tensors as Maps}

There is an equivalent, coordinate independent
view. A $p$-tensor is a multi-linear map
\[ T : \ub{\RR^n \times \cdots \times \RR^n}_{p}
\to \RR \]
such that
\[ T(\bf{a}, \bf{b}, \cdots, \bf{c}) = T_{i_1
\cdots i_p} a_{i_1} b_{i_2} c_{i_p} \]
(multi-linear = linear in each entry seperately).
\myskip
The tensor transformation rule ensures that the
map is independent of the choice of basis.
\begin{align*}
    T(\bf{a}, \bf{b}, \cdots \bf{c})
    &= T_{i_1 \cdots i_p}' a_{i_1}' b_{i_2}'
    \cdots c_{i_p}' \\
    &= (R_{i_1 j_1} \cdots R_{i_p j_p}) T_{j_1
    \cdots j_p} \times (R_{i_1 k_1} a_{k_1})
    \cdots (R_{i_p k_p} c_{k_p}) \\
    &= T_{j_1 \cdots j_p} a_{j_1} b_{j_2} \cdots
    c_{j_p}
\end{align*}
Alternatively, we can think of a tensor as a map
between lower rank tensors. For example, a
$p$-tensor can be viewed as a map
\[ T : \ub{\RR^n \times \cdots \times \RR^n}_{p -
1} \to \RR^n \]
The map is
\[ a_i = T_{ij_1 \cdots j_{p - 1}} b_{j_1} \cdots
c_{j_{p - 1}} \]
This is the way that tensors originally appear in
maths and physics, typically as a map from vectors
to vectors.
\[ \bf{u} = T \bf{v}  \implies u_i T_{ij} v_j \]
$T$ is a matrix but, importantly, transforms as a
tensor so the equation holds in all bases
\[ T_{ij}' = R_{ik}R_{jl}T_{kl} \]
or
\[ T' = RTR^\top \]

\subsubsection*{Tensor Operations}

\begin{itemize}
    \item If $S$, $T$ are tensors of the same
        rank, then so is $S + T$ and $\lambda T$
        for $\lambda \in \RR$.
    \item If $S$ is a $p$-tensor and $T$ is a
        $q$-tensor then we can form a $(p +
        q)$-tensor known as the tensor product
        \[ (S \otimes T)_{i_1 \cdots i_p j_1
        \cdots j_q} = S_{i_1 \cdots i_p} T_{j_1
        \cdots j_q} \]
        for example, given two vectors $\bf{a}$
        and $\bf{b}$ we can form the matrix
        \[ (\bf{a} \otimes \bf{b})_{ij} = a_i b_j \]
    \item If $T$ is a $p$-tensor then we can
        construct a $(p - 2)$-tensor by
        \emph{contraction}:
        \[ \delta_{ij} T_{ij k_1 \cdots k_{p - 2}}
        = T_{iik_1 \cdots k_{p - 2}} \]
        for example
        \[ T_rT = T_{ii} \]
        for a 2-tensor.
\end{itemize}

\myskip
We can combine the tensor product and contraction.
If $P$ is a $p$-tensor and $Q$ is a $q$-tensor, we
can form a $(p + q - 2)$-tensor. For example,
contraction on the first index gives
\[ P_{i k_1 \cdots k_{p - 1}} Q_{il_1 \cdots l_{q
- 1}} \]
for example given vectors $\bf{a}, \bf{b}$,
\[ \delta_{ij} a_i b_j = \bf{a} \cdots \bf{b} \]
is a zero-tensor. This is just the usual
inner-product. Another example is matrix
multiplication.