Harald Kirsch

genug Unfug.

$\def\v#1{\mathfrak{#1}} \def\vx{\v{x}} \def\vy{\v{y}} \def\vz{\v{z}} \def\mA#1#2{a_{#1 #2}} \def\ma#1#2{a^{#1}_{#2}} \def\t#1{\tilde #1} \def\tx{\t{x}} \def\ty{\t{y}} \def\d#1{\partial #1} \def\dd#1{\partial_{#1}} \def\pderiv#1{\frac{\partial}{\partial #1}} $


Contravariant, Covariant, Tensor

(III) Index Notation

If you were reading the previous two parts of this series in the hope to see indexes hop up and down between subscript and superscript, you may be disappointed. But don't dispair. Now that we understand that there exist two different types of basis-dependent $n$-tupels, it is time to talk about superscript indexes to distinguish contravariant tupels from covariant ones.

For the notation please refer to the previous blog post. To summarize, we have seen three transformations between vector space bases, \begin{equation*} \vy_j = \sum_{i=1}^n \mA{i}{j}\vx_i, \qquad x_i = \sum_{j=1}^n \mA{i}{j} y_j, \qquad \ty_j = \sum_{i=1}^n \mA{i}{j} \tx_i \end{equation*} where

  • the first is the basis vector transformation serving as a reference for the direction of the transformations,
  • the second is the transformation of coordinates of some vector $\vz\in V$, called contravariant because it goes in the opposite direction of the first, and
  • the third is the transformation of the values $\tx_i:=f(\vx_i)$ and $\ty_j:=f(\vy_j)$ of a linear form $f:V\to K$, called covariant because it goes into the same direction as the reference.

So there are basis-dependent $n$-tupels which are covariant and others which are contravariant. The simple idea is to distinguish them by raising the index for contravariant tuples to a a superscript. According to this rule, we must raise the index of coordinate values and from now on write them as $x^i$ and $y^i$ such that a vector $\vz$ now is written as \begin{equation} \vz = \sum_{i=1}^n x^i\vx_i = \sum_{j=1}^n y^j\vy_j . \label{eq:z} \end{equation}

And that was all? Not quite! Remember that for a fixed $j$ the $\mA{i}{j}$ are actually the coordinates of $\vy_j$ with regard to basis $\v{X}$. This means we must write, from now on $\ma{i}{j}$. Our transformations are then \begin{equation*} \vy_j = \sum_{i=1}^n \ma{i}{j}\vx_i, \qquad x^i = \sum_{j=1}^n \ma{i}{j} y^j, \qquad \ty_j= \sum_{i=1}^n \ma{i}{j} \tx_i \end{equation*} Strikingly, these three sums as well as the one in \eqref{eq:z} always zip up an upper with a lower index. And if this is the case then, as propsed by Einstein, the summation sign shall be left out. Having matching pairs of upper and lower index is enough to let us know that there is a sum over this index. In this Einstein notation, our three transformations can now be written as \begin{equation*} \vy_j = \ma{i}{j}\vx_i, \qquad x^i = \ma{i}{j} y^j, \qquad \ty_j= \ma{i}{j} \tx_i. \end{equation*} Very concise. This reminds me of programming languages, where some, like Java, are more verbose than others, like Scala or Perl. The tradeoff is that the less verbose a notation or language is, the more you need to know by heart. For the expert, verbosity is less efficient, while the beginner or even someone who has not used the notation for some time, may easily get lost.

But since the Einstein notation is so common I will use it too in the parts to come.