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I am trying to understand the meanings of "covariant transformation" and "contravariant transformation" and how they are related. I have read the related Wikipedia article and still feel I cannot state, with mathematical precision, the definition of these terms.

The Wikipedia article states that a covariant transformation, in the context of a vector space, is one that "describes new basis vectors in terms of old basis vectors". This is not a satisfactory definition unless, of course, no other transformations can be described as "covariant". I have seen however the word "covariant" being used to describe other sorts of transformations as "covariant". Namely, the "physicists" definition of co/contravariant transformations where components transform as such-and-such (which makes absolutely zero mathematical sense to me). This leads one to believe that co/contravariant transformations are always defined in terms of derivatives of coordinate changes and I don't believe this is the case.

I understand what co/contra-variant tensors are, at least from a mathematical perspective, so this is not a question about the meanings of "contravariant tensor" or "covariant tensor"; indeed, These concepts have been well-explained here.

My question then, in summary, What are lucid, self-contained and mathematically precise definitions of "covariant transformation" and "contravariant transformation"? A reference to such definitions would also work wonderfully.

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I don't think that "contravariant transformation" is established terminology in physics.

The problem with "covariant" is that in physics, this has a wide range of meanings, starting with "involving no unatural choices" up to the definition one sees in differential geometry motivated by general relativity, which is:

For a smooth real Riemann manifold $M$, a tensor $T$ of rank $\frac{n}{m}$ is a linear function which takes n 1-forms and m tangent vectors as input. When you choose a coordinate chart and dual bases on the cotangential space $d x_n$ and on the tangential space $\partial_n$ with respect to this chart, then the tensor has coordinate functions of the form

$$ T^{\alpha, \beta, ...}_{\gamma, \delta,...} = T(d x_{\alpha}, d x_{\beta}..., \partial_{\gamma}, \partial_{\delta}...) $$

With respect to these bases, a downstairs index is called covariant, an upstairs index is called contravariant. Now, a "covariant equation" or "covariant operation" is one that does not change its form on a coordinate change, which means that if you change coordinates and apply the choordinate change to all covariant and contravariant indices of every tensor in your equation, then you have to get the same equation, but with "indices with respect to the new coordinates".

A simple example would be: $$ T^{\alpha}_{\alpha} = 0 $$ with the Einstein summation convention: When the same index is used for a covariant and a contravariant index, it is understood that one should sum over all indices of a pair of dual bases.

Physicists would say that this equation is "covariant" because it has the same form in every coordinate chart, i.e. when I apply a diffeomorphism I get

$$ T^{\alpha'}_{\alpha'} = 0 $$ with respect to the new coordinates. Note that since we talk about general relativity, the kind of transformations are implicitly fixed to be changes of charts on a smooth real manifold. As I said before, when physicists talk about different theories, they may implicitly talk about other kinds of transformations. (Maybe you ran into some physicists who said "covariant transformation" when they meant "coordinate change", but me personally, I have not encountered this use of language.)

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As far as I can tell from the Wikipedia article, "covariant transformation" just means "the induced action of a linear transformation $f : V \to V$ on covariant tensors" and "contravariant transformation" just means "the induced action of a linear transformation $f : V \to V$ on contravariant tensors."

But apparently a covariant tensor is an element of $(V^{\ast})^{\otimes n}$ for some $n$ and a contravariant tensor is an element of $V^{\otimes n}$ for some $n$, which seems sort of backwards to me. Maybe I should say that a covariant tensor is a multilinear function $V^n \to k$ and a contravariant tensor is a multilinear function $(V^{\ast})^{\otimes n} \to k$.

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If I remember correctly (which I may not), the first volume of Spivak's Differential Geometry comments that the usage of "contravariant" and "covariant" in reference to tensors is counterintuitive to modern mathematicians, but maintained for historical reasons. –  Charles Staats Jun 7 '11 at 17:18
    
I personally regarded indices as being covariant or contravariant, rather than the objects themselves. An upper index is contravariant, and a lower index is covariant. Thus, the components of a vector $\mathbf{v}$ with respect to a basis $\{ \mathbf{e}_a \}$ are written with contravariant indices as $v^a$, and $\mathbf{v} = v^a \mathbf{e}_a$. Notice that the basis vectors are written with a covariant index, and that the contraction is invariant. (I guess you could say that the basis vectors are covariant, and the components of an invariant vector are contravariant.) –  Zhen Lin Jun 7 '11 at 17:55

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