What are the implications of representing basis vectors as directional derivatives?

Question

I am pretty new to differential geometry and general relativity. The notion of using directional derivative operators to represent basis vectors is starting to make sense to me, but I am still trying to get a deeper understanding of this notion. My question has four parts.

1. I believe that the mapping between basis vectors and directional derivative operators is specifically an isomorphism. This means that the mapping between them is "structure-preserving". What "structure" is preserved here, exactly?
2. What implications does this mapping being an isomorphism have for how literally we can take it that vectors are derivative operators? Can we rewrite vector/tensor equations as differential equations and then solve them to get a physical result?
3. Perhaps there are other ways we could have expressed vectors, other than as directional derivatives, that was dependent solely on the structure of the manifold that they are tied to. Is there a geometric interpretation to representing basis vectors specifically as derivative operators? Or is the usage of derivative operators not meant to be taken that strictly?
4. Is it accurate to state that a directional derivative operator is what we mean by a vector?

Let me know if my question needs to be more specific. I am having trouble accurately expressing my confusion.

Answer 1

Math perspective here.

1. In differential geometry, the tangent space $T_p M$ at a point to a differentiable manifold (like spacetime) may literally defined to be the $\mathbb{R}$ vector space of derivations on local differentiable functions. This is the answer to your 1st question in that the isomorphism you are thinking about is an invertible linear map given by the fact that the set of coordinate derivations arising from any coordinates system $x^i$ given by $$\{\frac{\partial}{\partial x^i} \}_{i = 1}^n$$ is a basis for this vector space $T_p M$ (its a theorem that the above are always a basis). So the invertible linear map is sending an arbitrary derivation to its $n$ coordinates in the above basis. There's nothing more complicated here than the fact that a basis for an abstract vector space allows it to be isomorphically related to $\mathbb{R}^n$; vector space structure is being preserved.

2. As stated above, tangent vectors are in every way directional derivatives (more properly derivations on basically the ring of smooth functions near $p \in M$). So if in a computation a vector appears acting on a function on the manifold, just use regular calculus: essentially one can get differential equations this way.

3. No there is not. Tangent vectors must be differential operators in differential geometry and this is quite geometric actually. If you think about it, geometry is about expressing ideas of calculus that work without any specific coordinates - so democratically across all coordinates. So what abstract idea might a tangent vector express? Well, humanity has decided to answer that question by thinking about what should it be able to do. Measuring the infinitesimal rate of change of a function in its direction is the answer. In this thread of thought, it helps to take on the relative point of view of mathematics, to define a structure: think about how it should transform. That is to say, if we have an appropriate notion of tangent space to a manifold, and a structure preserving map $f: M \to N$ (a smooth map of smooth manifolds), one should get an linear map at the level of the tangent spaces - again a structure preserving map on the dependent new object. I.e. one gets at each $p \in M$ the tangent map $$df_p : T_p M \to T_{f(p)} N$$ Since manifolds are locally modeled by $\mathbb{R}^n$, trying to define the tangent map in the local model as the simple Jacobian matrix of the map $f$ in local coordinates forces the simple numbers one should wish the represent the components of the tangent vectors (their existence in the local model) to transform like directional derivatives so that the tangent map is well-defined across all coordinates. Thus you discover that the abstract notion of a tangent vector must be a directional derivative footnote¹

4. Certainly.

Let me give you a very important example. Let $\gamma : I \to M$ be a curve (worldline), that is to say a smooth map of some interval in the real line (proper time) to spacetime ($M$). As you expect, the tangent vector to the curve at time $t$ is the best possible linear approximation to the curve there. So it ought to be an element of $T_{\gamma(t)} M$ as the image of the curve exists within spacetime. Well which element? As discussed above when we have a map from one manifold to another (proper time manifold to spacetime), we can naturally "pushforward" tangent vectors along it, giving the "best possible" version of the tangent to $I$ as something tangent to $M$. So how can we make a derivative of a function on proper time a derivative of a function on spacetime? Let $\partial_t$ be the tangent vector to the time interval $I$ at time $t$ (abuse of notation). The answer is this way. The pushforward derives functions $f$ on spacetime as

$$(d\gamma_{t} (\partial_t)) f := \partial_t (f \circ \gamma)$$ By inserting the identity via the coordinate system with $x^{-1} \circ x$ in between $f$ and $\gamma$ in the above, you discover via the multivariate chain rule that the above expression is $$\sum_{i =1}^n \frac{d (x \circ \gamma)^i}{d t}\big|_t \frac{\partial (f \circ x^{-1})}{\partial x^i}$$ From which the underlying derivation may be seen to be $$\sum_{i =1}^n \frac{d (x \circ \gamma)^i}{d t} \big|_t \frac{\partial}{\partial x^i}$$ following a common abuse of notation which really has to do with the above the theorem about basis of space of derivations and an isomorphism of rings (essentially smooth functions on the manifold and in the local model are the same thing; I certainly hope so, this again is the manifold principle).

What's the big takeaway? Don't think too hard about this; taking a course in differential geometry won't even tell you much more at the end of the day than tangent vectors to a differentiable manifold are derivations.

^{1 _{In mathematics this concept is called functorality and comes from the land of category theory}}