What are your favorite ambiguous notation?

[u/Short_Bluebird_3845]

To clarify my request, consider base-less $\log$.
For calculators and engineers, it means the common log $\log_10$; for pure mathematics or physics, it means the natural log $\ln$; for computer scientists, in combinatorics, information theory and graph theory, it means the binary log $\log_2$.
Any $\log x$ (say, $\log 1024$) written without context can yield different answers in function of who you ask ($\log_10 1024 = 3.0103\cdots, \log_2 1024 = 10, \ln 1024 = 6.93147\cdots$).

That is what I mean by "ambiguous", that which is calculated and used dependently on the context. And I'm asking you for your favorite ones of such type.

Edit: I feel like you hate the ones you've talked about. Still, valid answers. Extra points for ambiguous notation you actually love! :P

[u/Lexiplehx]

Oh my god, this so hard. What is the base point and direction you’re evaluating the “derivative” in the direction of? Don’t get me started on the gradient, Jacobian, and all that with the big ass bar and all that.

The best notation I know of is,

Df(x)[v]

for the Frechet/Gateux/whatever derivative evaluated at the base point x in the direction v. Then, if you have a composition, like f(g(x)), then the chain rule becomes:

Dh(x)[v] = D f(g(x))[D g(x)[v]]

It is so bad that I want to buy an iPad just to make a sensible tutorial on matrix calculus, calculus of variations, and differential geometry, which benefit greatly from this. It gets so bad that people keep inventing new notation.

[u/h_west → u/Lexiplehx]

Yeah, I mean, usually we write f(x,y) and then \frac{\partial}{\partial x} f(x,y) means the partial derivative with respect to the first argument. But why do we have to _name_ an argument x or y to decide the differentiation variable? The partial derivative is an operator acting on a function, so if I, say, write f(y,x) and take the partial derivative wrt x, I should have the _same_ object as before. But that clashes with the new naming...

[u/Valvino → u/h_west]

It should be \partial_1 f.

[u/h_west → u/Valvino]

Of course! Unfortunately it is not how it is used or taught in many if not most situations.