There are a number of posts on this site asking similar questions and some of them have been answered (to my taste) at least partially but none give a complete answer that I am satisfied with. See links at the bottom of this question for a small selection of posts asking related (or even the same) questions.
My question is as follows. The following is often written down:
$$
\nabla = \frac{\partial}{\partial_x} \hat{x} + \frac{\partial}{\partial y}\hat{y} + \frac{\partial}{\partial z} \hat{z}
$$
Some people will call this an operator, some will call it a vector, some will call it a vector operator, and some will adamantly claim that it is not properly anything at all and you shouldn't call it any of these things and you should just treat it as a "notational convenience".
One can then go on to use this "vector operator" to calculate things like $\nabla f$, $\nabla \cdot \vec{F}$ or $\nabla \times\vec{F}$ where the operator is treated notationally as if it were a vector.
First I want to take issue with the final claim that it is purely a notational convenience. I think it is more than just a notational convenience for the following reason. It is possible, by following certain transformation rules, to express $\nabla$ in different coordinate systems, for example cylindrical or spherical. That might be fine, but there is a FURTHER point which makes me think $\nabla$ must be more than a notational convenience. if you express $\nabla$ in different coordinates you can then calculate something like $\nabla \cdot \vec{F}$ in the new coordinates and get the right answer. An answer which you could have arrived at by explicitly converting the cartesian expression for $\nabla \cdot \vec{F}$ into the new coordinate system. In other words, the $\nabla$ allows you to actually skip a step of calculation you would have had to do otherwise. This is evidence that the symbol carries some sort of mathematical structure to it which should be able to be captured in an independent definition.
To that end I'm interested in a coordinate free definition of this symbol. The definition I gave above relies on using the usual Cartesian coordinates above. I have searched but haven't been able to find a coordinate free definition of the $\nabla$ symbol. Can one exist? In particular, I am interested in such a formula so that it is algebraically evident how one should calculate the components of $\nabla$ in any given coordinate system.
Is there a coordinate free definition of $\nabla$?
I am aware of a few complications with this endeavor that I'll just list here:
1) If this is to be some kind of vector or some kind of operator then it is not clear what space it should live in. For example, it is an object which can take a function $f$ and map it to a vector space. But at the same time it is an object which can be fed as an argument to a dot product together with a vector (form a different space) and return a scalar.
2) If I put on my differential geometry hat it becomes a very weird object. In differential geometry I come to think of vectors as actually being things like $\frac{\partial}{\partial x}$ and that $\vec{x}$ notation is eschewed. However the $\nabla$ symbol above contains both of these sitting next to each other. it's like a vector of vectors.. The idea of two vectors sitting next to eachother made me think it might be some kind of rank 2 contravariant tensor but I think that may have been a stretch.
3) I am aware that the cross product and curl operator are only defined in 3 dimensions so it does not need to be pointed out that that limits the possibility of defining such an operator for arbitrary dimension. I am happy to say we are working in 3 dimension.
4) I understand that the idea of divergence and curl depends on the presence of a metric for a space. Ok, that is fine. We can work in a space that has a metric defined on it.
5) Maybe the metric needs to be flat? Even that is fine as long as we can work in coordinate systems such as cylindrical or spherical where the metric is still flat but no longer has a trivial component representation. I am happy to restrict analysis to $\mathbb{R}^3$ if that is necessary.
6) Finally if such a definition truly cannot be formulated then could you at least answer why I can calculate BOTH $\nabla f$ and $\nabla \cdot \vec{F}$ by either 1) computing $\nabla f$ or $\nabla \cdot \vec{F}$ in xyz coordinates, then convert everything to spherical or 2) compute $\nabla$ in xyz coordinates, covert to spherical, then calculate $\nabla f$ and $\nabla \cdot \vec{F}$ and get the same answer in both cases? It just seems slightly too powerful/structured to be JUST a notational convenience.
Here are a few other related questions:
Is there a general formula for the del operator $\nabla$ in different coordinate systems?
Best Answer
Looking at the coordinate-free expressions that you've found in Wikipedia, it is easy to convince yourself that $\mathrm{grad}$, $\mathrm{curl}$, and $\mathrm{div}$ are instances of the exterior derivative in disguise: just think of the musical isomorphisms and the Hodge star as of means of identification. In the same article one can find a coordinate-free formula, that can be taken as the definition of the exterior derivative. This observation essentially closes the question.
The classical vector calculus deals with $\mathbb{R}^3$, which possesses some specific or exceptional structures, in particular, it has a canonical (Euclidean) coordinate system, the Euclidean metric, and the cross product, that all are extensively used in theory and calculations. If you want tor restict yourself to this case, then I doubt that it is ever possible to find a pure coordinate-free way of expressing the quantities under consideration (i.e. $\nabla f$, $\nabla \cdot \vec{F}$, and $\nabla \times\vec{F}$), as the space $\mathbb{R}^3$ itself is defined by explicitly presenting a single coordinate chart! In other words, you are forced to deal with coordinates and the dimension-related tricks in order to handle these quantities.
Coming back to the expressions in Wikipedia, notice that they use the Hodge star, but we have not received yet any convincing answer on how to give a coordinate-free definition for it. This doubles my pessimism, but I can be wrong and overlook something important.
Nevertheless, I find that this question and the other attempts to answer it are very insightful. For further discussion I suggest to look at the references below.
The best picture that shows that $\mathrm{grad}$, $\mathrm{curl}$, and $\mathrm{div}$ are closely related is given in [1], where they are combined into the de Rham complex. This text is perhaps too advanced, but a diligent undergraduate should be able to follow the first two paragraphs there, and the details can be recovered from [2] and [3].
References: