Proposal: Add the tag grad-curl-div; make gradient, curl and divergence aliases thereof.
Edit 2: I've revised the proposal statement to Alexander Gruber's (in my view superior) suggestion in the comments. Implementing this proposal poses some advantages over the status quo:
- It would (productively) free the > 6k questions about the gradient from the rather broad tag vector-analysis.
- It would resolve the weird asymmetry of the tagging situations of the three operators.
- It would reflect the close pedagogical relationship of the three operators, as Alexander pointed out in the comments: In practice many questions about one operators at least implicitly involve one of the others.
- It should in principle resolve the persistent problem of users erroneously using the divergence tag for questions about convergence and divergence in the analytic sense.
Proposed tag-excerpt:
The gradient, curl, and divergence are first-order differential operators that play a fundamental role in vector calculus and its generalizations.
Proposed tag-wiki:
In vector calculus and in differential geometry, the gradient, curl, and divergence are fundamental first-order differential operators.
- The gradient acts on differentiable (scalar) functions, producing vector fields that encode the direction and magnitude of maximum increase of the function. It can be regarded as a special case of the Jacobian and of the covariant derivative.
- The curl acts on vector fields and yields vector fields that measure the direction and magnitude of rotation. Unlike the other two operators, the curl is only defined in $3$-dimensions, but it has an analog in $2$ dimensions that is sometimes also called the curl.
- The divergence acts on vector fields and produces functions that measure the quantity of the fields' source at each point.
These operators have been fruitfully generalized to broader settings. For example, generalizations of the gradient appear in distribution theory and functional analysis, and their applications include the method of gradient ascent (descent) in optimization theory.
Edit After the discussion in the comments here and in the Tagging chat room, I went to create this tag, only to find that it was assigned as a synonym for vector-analysis in 2014. That same discussion comprises an argument for de-synonymizing gradient, i.e., letting it function as an independent tag. The topic is certainly broad enough: ~6.5k questions mention the gradient; ~4.4k questions tagged with vector-analysis. Barring this, it would only be consistent to make curl and divergence synonyms of vector-analysis, too.
This tag is intended for questions about or involving the gradient operator, which is a major theme in vector calculus and also important differential geometry. Probably most questions using this tag would also be tagged with vector-analysis or vector-calculus, but this operation is particularly important and arises commonly in such questions.
The tag-excerpt would read something like:
The gradient is a first-order differential operator that measures that rate and direction of fastest increase of a differentiable function.
The tag-wiki would read something like:
In vector calculus and in differential geometry, the gradient is a differential operator generalizing the derivative that acts on differentiable (scalar) functions, producing vector fields. The gradient of a function at a point is a vector that encodes the direction in which the function increases the most rapidly as well as the rate of increase; as such the gradient of a function is a special case of the Jacobian. The gradient has been fruitfully generalized to distribution theory and functional analysis, and applications include the method of gradient ascent (descent) in optimization theory.
Some model questions for this tag could include:
(At the moment the first of these is tagged with gradient-flows, which is not appropriate, since the question is not asking about the gradient flow o.d.e., and that choice of tag probably reflects the gap that the proposed tag aims to fill.)
The analogous differential operators in ($3$D) vector calculus already have their own tag:
suggesting that a gradient tag is not too granular. (NB glancing at the search results show that the divergence is frequently misapplied, to questions about analytic convergence/divergence. Cf. Arnaud D.'s helpful comment.) Naive searches of questions for the three terms give $\sim6.5$k results containing $\texttt{gradient}$, which suggests wide applicability.
By contrast, question searches find
(Again, the latter of these includes many questions about divergence in the analytic sense.) These figures also suggest that if we do not add a gradient tag, then for consistency curl and divergence should be removed, thought that change would entail a loss of usability in my view.
When I first raised the proposal in the Tagging chat room, Martin Sleziak helpfully pointed out that the tag was previously used in 2013 but then quickly removed in an edit.