mwane

Well, is f really a function of x and y when only one of them can be chosen? Seems like an abuse of notation that introduces unnecessary special cases. Abstractions (here: a function) should be simple and well-defined, not overloaded by natural language.

I get that, but ultimately, why does the notation distinguish between the two classes "function with zero arguments held constant" and "function with n >= 1 arguments held constant"? It seems somewhat of an arbitrary distinction, when in fact the value of n is always clear from the context.

Not really. It's simply a general observation I made after reading your post.

Which means there is no real difference; if all the types are known, it's just some form of Hungarian notation?

I think a major problem with mathematical language is lack of discoverability. If I don't understand a specific symbol or notation, there is no way to click "go to definition" or look it up on Google.

Furthermore while mathematicians pride themselves on rigor, I would bet that a lot of proofs require the reader to fill in incomplete steps and/or contain notational mistakes. Just try to imagine what kind of code you would end up with if you did not have access to a compiler and relied only on your logical reasoning for writing correct code. Even if the code was reviewed by other peers, who themselves do not have access to a compiler, they would miss some mistakes because, being familiar with the subject, their brain would fill in the correct meanings.

What is the difference between d and del?

> but we know del F/del y = d/dx(del F/del y')

????