actually, i quite agree, that the way most books start with covariant versus contravariant is misleading. in fact, i would go one further and say that the approach is wrong!
many beginning books start with the transformation properties of coordinates, and call anything that follows the same transformation rules "contravariant".
then they show that the partial derivatives with respect to the coordinates follow the opposite kind of transformation law.
there are a couple of things wrong with this:
1. it is very coordinate dependent. these are geometric objects, independent of coordinates, and the way they transform under coordinate transformations is a consequence of what type of geometric object they are, not vice versa.
2. manifold coordinates are components of a vector. partial derivatives are basis vectors. comparing their transformation properties is like comparing apples and oranges.
3. in category theory, a functor is called contravariant if it reverses the direction of the morphisms, and covariant if it does not.
consider the category of smooth manifolds, and tangent bundles and of cotangent bundles. given a diffeomorphism between two manifolds, this is a morphism in the category Smooth. Pushforward is a functor from Smooth to Tan, and Pullback is a functor from Smooth to Cot, and it is easily checked that Pushforward is covariant, while Pullback is contravariant.
the pullback of a diffeomorphism from M to N is a map from the cotangent bundle of N to that of M. notice how the order of the N and M in that sentence got reversed.
so vectors are naturally covariant objects while covectors are naturally contravariant objects. the problem is, physicists have this notation exactly backwards, and for no reason! if you make your coordinates have raised indices, then the components of a vector have also raised indices. physicists call these contravariant vectors, but i am claiming that they are covariant.
Any books recommended for dummies?
I am also a beginner on tensor. And in fact, I know that there are two approaches to define or develop the tensor: index-free approach and convectional approach based on indices. In my point of view, the index-free one is more appreciated by math student. And later one is mostly introducted to engineering and physics student. There are lot of material on the website you can search, and there are two suggestions:
2) Introduction to Tensor Calculus for Gerenal Relativity by MIT
BTW, the index-free approach, indeed, explores deeper geometrical insight, but it requires more mathematical background. And I don't think free-index approach is for dummies. (There maybe some lecturers disagree with that! )
We have no freedom, but we have choice. And I choose "NO WAR".
A brief summary of the math. A manifold is a smooth object that has a linear tangent space at every point, which together make up the "tangent bundle", a family of vector spaces.
Each vector space can be subjected to a variety of constructions yielding a new vector space, e.g. we can take various "tensor products" of our original vector space, forming a space of tensors. When these new spaces of tensors are bundled together with one at each point of the manifold, we get a new bundle, a tensor bundle. All tensor constructions have to do with linear or multilinear functions on the original vector space.
A choice of a tangent vector at each point of the manifold is called a vector field. A choice of tensor at each point is called a tensor field.
A tangent vector field transforms by the jacobian matrix of the change of coordinate map, hence in category language is "covariant", but as said above, differential geometers have this backwards and for historical reasons call these fields "contravariant". This is clearly discussed by Lethe above in post 3.
If f is a differentiable function on the manifold, it has a directional derivative in the direction of every tangent vector. Thus f assigns a number to each tangent vector. This assignment is called df, the differential of the function. This family of linear functions on tangent vectors is called a cotangent vector, and defines a covector field on the manifold. If A is any vector space the dual space of linear functions from A to the real numbers is called A^, or "A dual".
This field df transforms by the transpose of the jacobian matrix of the change of coordinates map, hence is truthfully contravariant, but unfortunately is called "covariant" in the language of classical differential geometry.
These vector and covector fields are "rank one" tensors it seems, from my brief perusal of some sites suggested locally.
Higher rank tensors apparently refer to multilinear tensors constructed from tangent vectors.
For example, given two vector spaces A,B, the family of all bilinear functions from the cartesian product
AxB to the real numbers, is a new space which is essentially the tensor product A^tensB^.
If we apply this construction to all the tangent spaces of a manifold, we get a new tensor bundle, and the Riemannian metric tensor which defines the length of a vector is a field of these tensors. I.e. the dot product of two tangent vectors is a bilinear function on pairs of tangent vectors with values which are numbers, hence at each point belongs to T^tensT^ where T is the tangent space.
When coordinates are introduced for tangent vectors, then one has corresponding coordinates for tensors, and all these change in complicated ways under coordinate transformations.
These coordinate transformations began to "wag the dog" at a certain point in classical differential geometry, and can make it quite a hard subject if unrelieved by some attention to what the objects mean. This notational approach seems to dominate the sources I have been referred from this site. Some respondents have even been unable to recognize my description of tensors as even remotely related to what they have learned from such "concept free" sources.
An example of the notational approach I recall are some highly complicated formulas due to Christoffel, so terrifying as to have been referred to as "them ******- awful formulas!" by hapless students. (possible profanity omitted here).
On the other hand in the modern approach as presented in Spivak's differential geometry book, they are revealed to measure only the extent that two one - parameter subgroups of a certain lie group fail to commute with each other, i.e. they are merely local coordinate expressions for AB-BA.
I confess I am clearly an amateur in this topic, but I think we are at least talking about the same subject.
Remark: As you probably noticed, vector fields are covariant, and covector fields are contravariant, even in the modern category terminology. For example in topology homology is covariant and cohomology is contravariant. This further confusion caused Peter Hilton to suggest long ago that cohomology be called instead contrahomology, but it did not catch on. At least the classical people were consistent in their wrong terminology, and called vectors "contravariant vectors", and they called covectors "covariant vectors".