I'm just starting to teach myself about covariant and contravariant vectors. With the little knowledge I've acquired so far I'm wondering if, for an ordinary Cartesian vector $\mathbf{V}$, it's OK to write $$\mathbf{V}=V^{x}\hat{e}_{x}+V^{y}\hat{e}_{y}+V^{z}\hat{e}_{z}$$ or is the correct notation $$\mathbf{V}=V_{x}\hat{e}_{x}+V_{y}\hat{e}_{y}+V_{z}\hat{e}_{z}$$

with subscripts, which seems to be they way they are written in my textbooks? Does it matter? I understand there's no difference between Cartesian covariant and contravariant vectors. I'm just curious why subscripted components tend to be used. The first version does sort of look “more balanced”, but what's the official line on this?

Thank you


Yes, it does matter.

However if all you will ever need is cartesian and the metric is the euclidian one, then in fact there is no difference.

But this is most likely not what you want, so I would strongly advise you to use the notation $$ \mathbf{V}=V^{x}\hat{e}_{x}+V^{y}\hat{e}_{y}+V^{z}\hat{e}_{z}=V^\mu \hat{e}_{\mu} $$ only for vectors in a vector space $X$. The other notation $$ \mathbf{V}=V_{x}\hat{e}_{x}+V_{y}\hat{e}_{y}+V_{z}\hat{e}_{z} $$ should be changed to (use the letter $V$ only if you know what is going on!) $$ \mathbf{W}=W_{x}\hat{e}^{x}+W_{y}\hat{e}^{y}+W_{z}\hat{e}^{z}=W_\mu \hat{e}^{\mu} $$ because usually a covariant vector is a vector in the dual space $\text{Hom}(X,\mathbb R)$ and the $\hat{e}^{\mu}$ denote the dual basis. If you have a non-degenerate scalar product (metric) $g:X\times X\rightarrow \mathbb R$, then contravariant and covariant vectors are identified via $$ X \rightarrow \text{Hom}(X,\mathbb R) \quad V \mapsto g(V,-) $$ but usually the components $V^\mu$ of $V=V^\mu \hat{e}_{\mu}$ and the components of $g(V,-)=g(V,-)_\mu \hat{e}^{\mu}$ are NOT equal. And there is - one - confusing convention: we (especially in physics) write $$ g(V,-)=g(V,-)_\mu \hat{e}^{\mu} = V_\mu \hat{e}^{\mu} $$ that is, we use the same letter $V$ (but place the index downstairs). But don't worry, it takes some time to understand this notational mess.

  • $\begingroup$ Thanks. So if I'm talking about ordinary Cartesian vectors I use the notation $\mathbf{W}=W_{x}\hat{e}^{x}+W_{y}\hat{e}^{y}+W_{z}\hat{e}^{z}=W_\mu \hat{e}^{\mu}$ with a downstairs index for the components and upstairs for the basis vectors? Afraid you lost me after that but no matter. Is it possible to explain in fairly easy language what the problem is with using V as the vector symbol? $\endgroup$ – Peter4075 Mar 12 '12 at 20:45
  • $\begingroup$ One further question. The commonest notation for ordinary Cartesian vectors seems to be (I might well be wrong here)$\mathbf{a}=a_{x}\hat{e}_{x}+a_{y}\hat{e}_{y}+a_{z}\hat{e}_{z}$ with downstairs indices on both the components and basis vectors. What are the pros and cons of this notation and your $\mathbf{W}=W_{x}\hat{e}^{x}+W_{y}\hat{e}^{y}+W_{z}\hat{e}^{z}=W_\mu \hat{e}^{\mu}$ notation? Sorry if I'm making heavy weather about this. $\endgroup$ – Peter4075 Mar 12 '12 at 21:03
  • $\begingroup$ Ahh, I know your problems - I had them for a long time. It is confusing and nothing can be done about that. Let me explain further: If you are only working with cartesian coordinates, it doesn't matter at all, a vector is a bunch of numbers - you can use any notation you like. For the downstairs/upstairs convention: do you know Einsteins summation convention? $\endgroup$ – Blah Mar 12 '12 at 22:47
  • $\begingroup$ Yes, I do. I think that's why it puzzled me why textbooks use $\mathbf{a}=a_{x}\hat{e}_{x}+a_{y}\hat{e}_{y}+a_{z}\hat{e}_{z}$ instead of $\mathbf{V}=V^{x}\hat{e}_{x}+V^{y}\hat{e}_{y}+V^{z}\hat{e}_{z}=V^\mu \hat{e}_{\mu}$ which, to my eyes, looks more balanced with upper and lower indices. $\endgroup$ – Peter4075 Mar 13 '12 at 6:19

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