What does it mean for a vector to be a derivative? I'm reading on Killing vectors and Killing vector fields, and one notion that keeps coming up is a derivative being a vector. For example, it's put here(Eq. 5.46) that in 2-dimensional Euclidean space, the "vector" $R_{\theta}=\partial / \partial \theta$ (derivative with respect to polar angle) is given by,
$$R_{\theta}=(-y,x)$$
This reminds me of the the generators of a lie algebra, but I'm not totally sure. Could you help me understand what this is?
 A: It merely means that there is a one-to-one correspondence between directions (geometric vectors) and directional derivatives.
We need not use notation that overtly mentions this correspondence if we don't want to.  You could denote the polar basis vector by $e_\theta$ for instance.
So what does it mean to say that $e_\theta = (-y,x)$?
The statement says that $e_\theta$ can be expressed as a function of Cartesian coordinates: $e_\theta(x,y) = (-y,x)$.
Furthermore, the ordered pair on the right signifies a linear combination of the Cartesian basis vectors.  So it really means this:
$$e_\theta(x,y) = -y e_x + x e_y$$
Compare with the following statement that is explicitly about coordinate derivatives:
$$\frac{\partial}{\partial \theta} = -y \frac{\partial}{\partial x} + x \frac{\partial}{\partial y}$$
A: In $\mathbb {R}^n$ there is a natural correspondence between tangent vectors at a point (i.e: points of the vector space $\mathbb {R}^n$ taken fixed, which some call "affine space")  and directional derivatives: derivatives of real-valued functions along the direction given by a certain vector. So, in a more generic situation, in the case of a manifold, for example, we DEFINE the tangent vector space as the vector space of all derivations of real-valued functions. That is, linear maps from the vector space of differentiable functions $f: M \rightarrow \mathbb{R}$ to $\mathbb {R}$ that satisfies the known product formula of calculus: $${X : C^k(M) \rightarrow \mathbb{R}.}$$ Further, if $\{x^i\}_{i=1}^n$ are the coordinates around a point, its a very simple theorem (that is found in any book of differential geometry) that the tangent vectors(in the sense above) $\{\frac {\partial}{\partial x^i}\}_{i=1}^n$ form a basis for this tangent space. 
