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Working in curvilinear coordinates, one can define basis vectors corresponding to those coordinates. In the figure below (taken from here), $\{\mathbf{g_i}\}$ are base vectors corresponding to the change in $\{\Theta_i\}$. These are called covariant base vectors. One can also define another set of base vectors, $\{\mathbf{g^i}\}$, such that $\mathbf{g^i} \ \mathbf{g_j} = \delta_{i}^j$. These are called the contravariant basis and a vector $\mathbf{v}$ can be written as $\mathbf{v} = v^i \mathbf{g}_i = v_i \mathbf{g}^i$.

Now I understand that the summation convention says that if an index appears as a subscript and a superscript, the summation is implicit. My question is whether there is any (other) practical reason for denoting the components in the covariant basis with upper indices and in the contravariant basis with lower indices. For instance, it would be much more natural to me to write $\mathbf{v} = v_i \mathbf{g}_i = v^i \mathbf{g}^i$ (where now $v_i$ denotes the component $v^i$ from the figure). This way the indices in the components would match the indices in their basis: in the basis with the upper index, components would be with upper index and vice versa. The norm and scalar product would work out to be the same since $\mathbf{u} \mathbf{v} = u_i \mathbf{g}_i v^j \mathbf{g}^j = u_i v^i$. Obviously, in this case indices appearing twice at the same level have to be summed.

I also understand that the co- and contravariant components transform differently. However, here I'm interested in the benefits of the upper-lower notation as opposed to a matching, upper-upper and lower-lower notation.

basis vectors

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    $\begingroup$ I think the upper-lower notation helps as a 'general check', although I know this isn't exactly how it works, its gives a rough guide, which is the upper and lower indices should cancel out to result in as many upper or lower free indices as in the L.H.S term. e.g. $ x^\mu = g^{\nu\mu}x_{\nu}$. The indices don't exactly work like that if we're being mathematically rigorous, but its a nice general sort of thing. $\endgroup$ – SamuraiMelon Apr 13 at 22:58
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I also understand that the co- and contravariant components transform differently. However, here I'm interested in the benefits of the upper-lower notation as opposed to a matching, upper-upper and lower-lower notation.

But that's precisely the point - objects with a lower index (whether they be basis vectors or components) transform in one way, while objects with an upper index transform the other way, such that their contractions remain invariant. In your suggested convention, index placement does not determine transformation behavior, whereas in the standard convention it does.

Of course, this is all conventional. If you want, you can write all indices upstairs, but write "covariant" indices with red ink and "contravariant" indices with blue ink. But there is good reasoning behind the existing convention, and that (along with the fact that it is essentially universal at this point) is enough for me not to worry about rocking the boat.

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The components of a vector and the basis vectors transform differently. The components of a vector transform contravariantly and the basis vectors transform covariantly, so writing

$$\vec V=v_i\hat e_i \tag{1}$$

is not just different notation, it is incorrect. The choice to label upstairs indices as contravariant and downstairs as covariant is arbitrary (as far as I know), much like defining positive and negative charge.

The reason contravariant and covariant indices have to be contracted together and not contravariant-contravariant or covariant-covariant contraction is because in the context of physics they produce "frame invariant" quantities. Different observers will disagree on some measurements, but they will always agree on quantities defined by tensor contraction. I have actually asked this question before and G.Smith was nice enough to write out this answer with an explicit example.

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  • $\begingroup$ thank you for the answer. Note however, that I am not talking about covariant-covariant or contravariant-contravariant contraction. I'm talking about denoting components so that they match the indices in the base vectors. Thus $v^i (new) = v_i (old)$ in my naive proposition. Then $\mathbf{v} =v^i (new) \mathbf{g^i} = v_i (old) \mathbf{g^i} $ $\endgroup$ – Botond Apr 13 at 20:52
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    $\begingroup$ You need to be able to distinguish between covariant and contravariant indices somehow, how can you distinguish the two in your new system? $\endgroup$ – Charlie Apr 13 at 20:54
  • $\begingroup$ I guess this is the point I can't appreciate enough: why do I need to distinguish between the covariant/contravariant indices? I've come across this formalism through differential geometry but right now I just don't see why people are fixated on the way components transform... There must be a good reason though. $\endgroup$ – Botond Apr 13 at 20:57
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    $\begingroup$ Because when you change basis you need to make sure you adjust your vector/covector components correctly. If you make no distinction you will end up with incorrect answers. For instance in special relativity when you change frame you are performing a basis transformation, to correctly transform 4-vector components between frames you have to know if they are transforming covariantly or contravariantly or you will just get wrong answers. $\endgroup$ – Charlie Apr 13 at 21:00

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