I'm confused about the difference between a contravariant and covariant vector. Some books and articles seem to say that there really is no difference, that a vector is a vector, and can be written in terms of contravariant components associated with a particular basis, or can be written in terms of covariant components associated with the dual basis of the original basis. In other words, it is the components that are contravariant or covariant. Yet, elsewhere I read that the gradient is a covariant vector, and that velocity is a contravariant vector, because of the way the components transform. No mention of the original basis or dual basis. There seems to be a real difference in the nature of these vectors, as opposed to simply the components and bases used to express them. I would very much appreciate any insight that anyone can give me on this.