# Why are contravariant vectors denoted with a superscript (and not a subscript)?

I wonder whether the choice of denoting contravariant/covariant vectors with a superscript/subscript is arbitrary (and could have been made the other way around), or whether there is a specific reason to it.

To me, the more "natural" quantities are the contravariant ones, like positions in space and measurements, which outside of the world of tensors are indexed using subscripts. My natural choice would have been to use subscripts for these contravariants quantities in tensor notation. So while introducing tensor notation, I wonder what reason may be forcing me to change about 90% of my indices from subscripts to superscripts.

• If your coordinates carried subscripts, your default derivative operator with respect to that coordinate would carry a superscript. I like subscripts for derivatives, it fits well with the fraction notation for derivatives. – ziggurism Jan 27 '16 at 19:36
• Are you specifically referring to higher-order differentials such as $\frac{\partial^2}{\partial x^2_i}$? In that case it makes sense, I agree - superscript indices would be confusing. – bers Jan 27 '16 at 19:39
• Even first order operators. I agree that superscripts conflict with exponents in higher order differential operators, but that will be the case no matter which variable you choose to get the raised index. – ziggurism Jan 27 '16 at 19:41

Just thinking in terms of real functions of one variable, using the chain rule, we can see that a differential $dx$ gets multiplied by $\phi'$ under a change of variables $x=\phi(x')$. $dx=\phi'\,dx'$. Similarly we see that the differential operator $d \over dx$ gets divided by $\phi'$: $\frac{d}{dx}=\frac{1}{\phi'}\frac{d}{dx'}$, which makes total sense since the $x$ is in the denominator of Leibniz's fractional notation for derivatives.

Let's call the things which get multiplied by $\phi'$ under a change of variables covariant and the things that get divided contravariant. Covariance is associated with being on top of the Leibniz notation, and contravariance is associated with being below.

This is why coordinates should have raised indices; they transform covariantly.

• My question was: why do things "that transform contravariantly" have "superscripts". Now you are explaining that things that "transform covariantly" have raised indices (superscripts). I am confused. – bers Jan 27 '16 at 22:40
• Yes, I realized as I finished it that my terms appear to contradict yours. I will edit to make it clearer. – ziggurism Jan 27 '16 at 22:57
• Thanks :) Please note that my main question is not "what is a contra/covariant thing", but "why do we use super/subscripts" do indicate that. – bers Jan 27 '16 at 22:59
• Yeah, I mean, I'm standing by my answer to that question. Leaving aside for the moment whether we call it "contravariant" or "covariant", the answer is: we used raised indices for things that transform like the numerator of a Leibniz-style fractional derivative notation, and lowered indices for things that transform like the denominator. – ziggurism Jan 27 '16 at 23:01
• Oh, I see. Thanks! – bers Jan 27 '16 at 23:04