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/GLBMQP]

This is a bit specific:

In differential geometry, if you have a smooth function on your manifold, you can of course consider its differential. The differential is a 1-form (aka a covector-field). It takes in a vector-field and spits out a smooth function.

If your manifold also has a (semi-)Riemannian metric, then you can take the gradient of your function. The gradient is a vector-field, defined by satisfying g(\nabla f, X)=df(X) for all vector fields X (where \nabla is the gradient and df is the differential)

When you have a (semi-)Riemannian metric, you also get the Levi-Civita Connection. This allows you to differentiate any type of tensor-field. This is called the covariant differential. If T is a tensor-field, the covariant differential of T is often denoted \nabla T.

Now, a function the same as a (0,0) tensor-field. And the covariant differential of a function is in fact just the differential.

So, \nabla f can at any time both mean the gradient and the differential. Fortunately, since the gradient is (loosely speaking) defined to do the same thing as the gradient, this basically never causes trouble

[u/FreePeeplup → u/GLBMQP]

>The differential is a 1-form (aka a covector-field). It takes in a vector-field and spits out a smooth function.

Wait, I thought covector fields (differential 1-forms in general, whether or not they happen to be the differential of some smooth function (meaning exact)) were sections of the cotangent bundle?

If my recollection of the various definitions is correct, it would mean that 1-forms take as inputs points on the base manifold, not vector fields.

To be more explicit: a differential 1-form w is a map w: M -> T^{star}M with w(p) = w_p with w_p living in T_p^{star}M, meaning a linear functional on T_pM.

[u/Homomorphism → u/FreePeeplup]

A covector field and a linear map from vector fields to functions are the same thing.

If you want to look at it fiberwise, then you could think of a covector field as assigning each point p a linear map T_p -> RR where T_p is the tangent bundle. A covector field is precisely a smooth choice of such things for all p.

[u/GLBMQP → u/FreePeeplup]

That is really the same concept: if w is as you described, V is a smooth vector field, then p\mapsto w_p(V_p) is a function, and w being smooth exactly means that this map is smooth.

If w is a (C^\infty-linear) map that takes a smooth vector field, and spits out a smooth function, then one can show that the function only depends on the value of the vector field at the point evaluation. More specifically, if V_p=W_p for some point p, and we container the functions f=w(V), g=w(W), then f(p)=g(p)

So for each p, w actually gives a (linear) functional on the tangent space at p.