I am trying to work through do Carmo's book. I have just gotten into his sections on abstract surfaces. I think I get the general idea: In the more familiar surfaces, we rely on $\mathbb{R}^3$ space to calculate various and sundry quantities like the metric. In abstract surfaces we do the opposite and define those quantities and then, when possible, derive the "nature" of the space from them. Then he starts talking about how a vector is actually a derivative. So far, so good, I think. But then he says we can define the familiar functions from $\mathbb{R}^3$ as inner products of the derivatives. For instance,
$ E(u,v) = \left < \frac{\partial}{\partial u},\frac{\partial}{\partial u}\right>$
And this is where I am totally lost. I have no idea what it would mean to define the "inner product of derivatives", let alone how to calculate such an inner product. $E(u,v)$, being a function, returns a number from any $u, v$. A derivative cannot at all be associated with a number in that same way unless it is also specified what function said derivative is acting on. Or that's what they always told me in calculus class. I am completely confused. Can anyone clear things up for me?