Contravariant and Covariant Vectors

Question

Are there contra variant and covariant vectors or are there only contra variant and covariant basis for vectors and these basis transform into further contra variant and covariant basis respectively according to certain co-ordinate transformation laws, keeping the vector as it is, which can be viewed as an arrow ?

Answer 1

A good analogy is that vectors behave like $n \times 1$ column matrices and covectors behave like $1 \times n$ row matrices. The analogy becomes even stronger when you realize the transpose operation happens to be the metric transpose, for the specific case where the metric is the dot product. (in particular, if you don't have a metric, then you really, really shouldn't be thinking of "column vectors" and "row vectors" as being the same thing)

The transformation laws can be seen easily in this picture too; a change of basis matrix $B$ transforms vectors like $v \mapsto Bv$ and covectors like $\omega \mapsto \omega B^{-1}$ (so that their product remains the same: $\omega v \mapsto \omega B^{-1} B v = \omega v$).

Going back to multivariable calculus, a vector function of one variable is a function whose values are $m \times 1$ matrices. However, the derivative of a scalar-valued function of $n$ variables is a $1 \times n$ matrix. Similarly, a vector-valued function of $n$ variables has a derivative that is an $m \times n$ matrix.

Note that the dot product isn't involved at all when computing a directional derivative of a scalar function: you can simply multiply the row vector $\mathrm{d}f$ by the column vector $v$ to get

$$ \nabla_v f = (\mathrm{d}f) v$$

The usual formula for the directional derivative in terms of the dot product is an artifact of mistakenly treating $\mathrm{d}f$ as another vector (rather than a covector), and so you have to use the dot product to undo that mistake.

Okay it's not really a mistake, per se: I imagine many would find multivariable calculus even more complicated if they were trying to learn the difference between vector and covector at the same time they're trying to learn calculus, so calculus texts just transpose all the covectors to become vectors to avoid the issue.

Of course, that comes at the cost of obscuring the issue when it comes time to actually learn it. I was lucky and figured out the row vector/column vector distinction on my own when I took calculus. Personally, I found it easier to keep everything straight in my mind when I did so.

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Incidentally, if you're looking for a graphical depiction of a covector, you need to imagine it in relation to how you think of functions, not points. A vector, in some sense, is something that tells you how to move from one point to another. A covector, however, is something that tells you how a function varies.

I, personally, like to think of functions algebraically rather than geometrically. e.g. the covector field that is the derivative of $x^2 y$ I like to see as the algebraic formula $2xy \mathrm{d}x + x^2 \mathrm{d} y$ rather than as some sort of geometric picture.

I've seen people who like to visualize functions by looking at their level curves (i.e. the curves defined by $f(\mathbf{x}) = a$ for various $a$), extend this visualization to covectors, by imagining them as an infinitesimal element of the level curve (in the same way one might imagine a vector as an infinitesimal element of a curve). Of course, in $n$ dimensions, we need the $(n-1)$-dimensional analogs, rather than curves.