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The notation for tensors is like the plumbing in a very old Vermont farmhouse. It may once have been intentionally designed, but after that it just evolved. As an example, it seems that depending on taste in notational systems, there may be as many as six flavors of rank-1 tensors or as few as one.

In the notation introduced by Sylvester in 1853, we have $\mathbf{x}=\sum x^j \mathbf{e}_j$ and $\boldsymbol{\omega}=\sum \omega_j \mathbf{e}^j$. The flavors are invariant vector $\mathbf{x}$, contravariant n-tuple of components $[x^j]$, covariant vector $\mathbf{e}_j$, invariant covector $\boldsymbol{\omega}$, covariant n-tuple of components $[\omega_j]$, and contravariant vector $\mathbf{e}^j$.

In the Penrose diagrammatic notation (similar to Cvitanovic's birdtracks and Peterson's trace diagrams), we arguably have only one flavor of rank-1 tensor, which is a vertex with one edge. (OK, I cheated. The edge is directed, so that does give us two flavors of diagrams, an innie and an outie. But if there's a metric we can reverse the arrowhead at will.)

My question: Throwing out all the historical baggage, how many of these distinctions are actually important enough that they deserve to be built into the notation? Let's assume we have a metric.

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If you've got a vector space ${\bf V}$ with a basis $\{e_i\}$, and you want to talk about an arbitrary vector, you can call it ${\bf x}$, or you can write it as $\sum \alpha_i e_i$ . If you also have a preferred isomorphism $j$ from ${\bf V}$ to the dual space ${\bf V}^\*$, you can also write your vector as $j({\bf y})$ for some ${\bf y}\in {\bf V}^\*$, or, if you're being sloppy, you can identify ${\bf x}$ with ${\bf y}$. Or you can write ${\bf y}=\beta_i f_i$ after picking a basis for the dual.

How many of these different notations do we "need", and why does it matter whether ${\bf V}$ happens to be the tangent (or cotangent) space to some manifold at some point?

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Although with a metric one can pretend a cotangent vector is a tangent vector and vice versa, my experience is that it is important to keep track of which is which and be conscious of when one uses the isomorphism. In particular, the two transform quite differently if you do a change of basis or co-ordinates, and this can matter. –  Deane Yang Feb 7 '13 at 13:52
Deane Yang: Agreed, of course --- but I still don't see where this has anything to do with tangents and cotangents. Whenever you choose an isomorphism between a vector space $V$ and its dual, it can be important to keep track of the way in which various bases for $V$ are mapped to bases for $V^∗$, and in particular to remember that when you change bases in $V$, the corresponding bases in $V^*$ transform in a different way. –  Steven Landsburg Feb 7 '13 at 22:43

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