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I have read some introductions to geometrical ideas and tensors and physics and what some of them do (see, for example, Frankel's Geometry of Physics) is define a vector as a first order differential operator on functions. I get how they do this: there is a natural isomorphism between the vector spaces $\mathbb{R}^n$ and the vector space $V =\{v\ \cdot \nabla :v\in \mathbb{R}^n \}$. What I don't get is why they do this. Even re-reading the parts where they describe it, they don't seem to motivate it very much.

In short, what do we "get" out of redefining vectors in this way?

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What we get out of this is the ability to define vectors where there is no obvious way to choose a vector from $\mathbb R^n$. In particular, it allows us to define the tangent space of a manifold without first embedding the manifold in a real vector space.

If we have a $k$-dimensional submanifold $M\subseteq\mathbb R^n$, then we can literally take the tangent vectors at a point $p\in M$ and form a $k$-dimensional vector space $T_pM$ out of these. This vector space will be a subspace of $\mathbb R^n$. We can also identify each of the vectors with a directional derivative at $p$: Let $f:M\to\mathbb R$ be a smooth function and $\gamma:[-1,1]\to M$ a smooth curve with $\gamma(0)=p,~\gamma'(0)=v$. Then $v$ is in $T_pM$ and we can identify the map $f\mapsto \frac{\mathrm d}{\mathrm dt}f\circ\gamma\large\vert_{t=0}$ with $v$. Since $\frac{\mathrm d}{\mathrm dt}f\circ\gamma\large\vert_{t=0}$ only depends on $\gamma(0)$ and $\gamma'(0)$, it doesn't matter which path in particular we choose. So we can say that this derivative is the directional derivative in direction $v$, regardless of what $\gamma$ we chose in particular. And we can do the same with any $v\in T_pM$, so we can really identify the tangent space $T_pM$ with the space of directional operators assigning to each function a specific directional derivative at $p$. Now if we take a manifold which is not embedded in Euclidean space, we can't do the first part. The "literal" tangent vectors don't have a surrounding space into which they could point, so we'd be really hard pressed to even say what those vectors should be. Of course, we could try embedding the manifold into some higher-dimensional space, but it's really arbitrary how we do that, and the resulting tangent vectors depend on how we embed the manifold. And definitions which depend on an arbitrary choice are generally something that is best avoided.

But we can still do the second part: The derivative of $f\circ \gamma$ for some smooth curve $\gamma$ in the manifold is well-defined. So for general manifolds, defining the tangent vectors can still be done via directional derivatives. So we do that.

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