2015-01-18

# Contravariant, Covariant, Tensor

## (II) Covariance

After I understood
where the term *contravariant* comes from, I am now ready
to explain *covariant*. As before we have a vector space
$V$ over a field $K$ with two bases
\begin{align*}
\v{X} &= (\vx_1,\dots,\vx_n), \qquad \vx_i\in V, \\
\v{Y} &= (\vy_1,\dots,\vy_n), \qquad \vy_j\in V
\end{align*}
and a set of $\mA{i}{j}\in K$ that transform $\v{X}$ into $\v{Y}$
according to
\begin{equation}
\vy_j= \sum_{i=1}^n \mA{i}{j}\vx_i .
\label{eq:vy}
\end{equation}
Further we look at a linear form $f:V\to K$, i.e. a function from
$V$ into $K$ that assigns an element $f(\vz)$ to each $\vz\in V$
and is linear. In particular $f$ provides us with two $n$-tupels
$\tx_i:=f(\vx_i) \in K$ and $\ty_j:=f(\vy_j) \in K$, one for each
of the bases.

This reminds of the coordinates of a vector $\vz\in V$ which
are also $n$-tupels of values depending on the selected basis, and
we can ask whether and how we transform the $\tx_i$ into the
$\ty_j$. But this is not difficult:
\begin{align*}
\ty_j &= f(\vy_j) \\
&= f\left(\sum_{i=1}^n \mA{i}{j}\vx_i\right)
& &&\text{by \eqref{eq:vy}} \\
&= \sum_{i=1}^n \mA{i}{j} f(\vx_i) \\
&= \sum_{i=1}^n \mA{i}{j} \tx_i.
\end{align*}
We see that the $\mA{i}{j}$ transform basis vectors $\vx_i$ into
$\vy_j$ (see \eqref{eq:vy}) as well as the coefficients $\tx_i$
into $\ty_j$, hence these coefficients of a linear form $f$
transform in the same direction as the bases and are therefore
**co**variant.

To summarize, the $\mA{i}{j}$ perform for us the following transformations:

- basis vector $\vx_i \longrightarrow \vy_j$ (reference)
- vector coordinates $y_j \longrightarrow x_i$ (contravariant, opposite direction of reference)
- linear form coefficients $\tx_i \longrightarrow \ty_j$ (covariant, same direction of reference)

If you were reading hoping to see indexes hop up and down between subscript and superscript, you may be disappointed, but don't dispair. I think that only now that we clearly understand that there exist two different types of basis-dependent $n$-tupels, it is time to talk about superscript indexes to differentiate contravariant tupels from covariant ones.

And the rules are simple. The components of a basis-dependent $n$-tupel have their index

- as a
*subscript*, like $\tx_i$ and the basis vectors $\vx_i$ itself, if the $n$-tupel transforms*covariant*, and - as a
*superscript*like $x^i$, if the $n$-tupel is*contravariant*.