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Sorry, I'm not an advanced abstract algebra person, so I haven't seen this situation addressed before. Suppose you abstract the differential operator to just any ol' linear operator $L$ over a field which has the conditions

$L[c_1 u + c_2 v] = c_1 L[u] + c_2L[v]$ and $L[u \cdot v] = v \cdot L[u] + u\cdot L[v]$ and $L[c_1] = 0$ for any constant $c_1.$

Then must such an operator be the differential operator when acting over any field over which such an operator exists?

Purely in the context of a field, you don't necessarily have limits so I don't know if I postulated this question properly, it may be the case that a field is required to be "continuous" in some sense like a Lie group for such an operator to be the differential operator. I think there are conditions over which field elements are unique though too. I never studied topology in-depth, but this may require an extra Hausdorff condition for infinite fields.

So, the answer may end up being "Yes but only over (such and such) a field, otherwise no and here's an example..."

StackQuest
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    Such a map is called a derivation. Most derivations on most algbras cannot be interpreted as derivatives. See https://en.wikipedia.org/wiki/Derivation_(differential_algebra) – GEdgar Jan 03 '22 at 00:08
  • Okay, why "can't" they? Is there an example of a non-derivative object where what you're saying occurs? – StackQuest Jan 03 '22 at 00:09
  • Here is a broad class of cases where they can be identified with a derivative-type object: https://en.wikipedia.org/wiki/Tangent_space#Definition_via_derivations – Randall Jan 03 '22 at 00:14
  • You can have more than one derivation operator on a field. For example, $\mathbb{R}(x,y)$ has two obvious ones; $\frac{\partial}{\partial x}$ and $\frac{\partial}{\partial y}$. – Mark Saving Jan 03 '22 at 00:33
  • Okay, that's interesting. Although this is unrelated, is there a related abstraction of the anti-derivative called an integration or integrator or anything like that? – StackQuest Jan 03 '22 at 00:36

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