A better notation for a differential?

[MichPod]

A $\frac{df}{dx}$ notation is problematic. Obviously, the letter 'd' has very different meaning when applied to the function or to the argument. Additionally, a separate letter '$\partial$' is used to denote a partial differential (a very rare case in math when a notation used for a general case of partial differential cannot be used for a more specific case of a full differential). Additionally, the $\frac{\partial f}{\partial x}$ expression is ambiguous as the arguments of the 'f' function are only implied and not stated explicitly. Additionally, a differential of a multivariable function is a covector, a second differential is a tensor, but that fact is just masked by considering the partial differentials separately. I guess, there may be more recognized disadvantages of the above notation.
The question: were there any successful attempts to improve the notation for the differential specifically in math? I mean not in the school textbooks, but generally. May be some advanced books and topics may use a different, better notation?

 

[fresh_42]

A $\frac{df}{dx}$ notation is problematic. Obviously, the letter 'd' has very different meaning when applied to the function or to the argument. Additionally, a separate letter '$\partial$' is used to denote a partial differential (a very rare case in math when a different notation is needed to be used for a general case of partial differential and a more specific case of the full differential). Additionally, the $\frac{\partial f}{\partial x}$ expression is ambiguous as the arguments of the 'f' function are only implied and not stated explicitly. Additionally, a differential of a multivariable function is a covector, a second differential is a tensor, but that fact is just masked by considering the partial differentials separately. I guess, there may be more recognized disadvantages of the above notation.
The question: were there any successful attempts to improve the notation for the differential specifically in math? I mean not in the school textbooks, but generally. May be some advanced books and topics may use a different, better notation?

Here is a (probably incomplete) list of what is already used:
$$D_{x_0}L_g(v)= (DL_g)_{x_0}(v) = \left.\frac{d}{d\,x}\right|_{x=x_0}\,L_g(x).v = J_{x_0}(L_g)(v)=J(L_g)(x_0;v)$$
Source https://www.physicsforums.com/insights/journey-manifold-su2mathbbc-part/

 

[pbuk]

Here is a (probably incomplete) list of what is already used:

Don't forget at the other end of the spectrum $y', \ddot x$ etc, and also $\nabla, \nabla \cdot, \nabla \times$.

I don't see that this is a big thing: in general one uses the simplest notation whose meaning is clear in its context.

 

[Mark44]

A $\frac{df}{dx}$ notation is problematic. Obviously, the letter 'd' has very different meaning when applied to the function or to the argument.

One can always be more explicit as to how the derivative function should be evaluated, like so:
$$\left. \frac{df(x)}{dx}\right|_{x = x_0}$$

Additionally, a separate letter '$\partial$' is used to denote a partial differential (a very rare case in math when a different notation is needed to be used for a general case of partial differential and a more specific case of the full differential). Additionally, the $\frac{\partial f}{\partial x}$ expression is ambiguous as the arguments of the 'f' function are only implied and not stated explicitly.

If this is a problem, the arguments can be made explicit:
$$\frac{\partial f(x, y)}{\partial x}$$
Or if the partial is to be evaluated at a particular point:
$$\left.\frac{\partial f(x, y)}{\partial x} \right|_{(x, y) = (x_0, y_0)}$$

The question: were there any successful attempts to improve the notation for the differential specifically in math? I mean not in the school textbooks, but generally. May be some advanced books and topics may use a different, better notation?

As @pbuk already noted, in the direction of more ambiguity, there are notations like $y'$ and $\ddot x$ that are more or less due to Newton. I view the Liebniz notation (i.e., df/dx etc.), as being a considerable improvement in many circumstances.

 

[martinbn]

The standard notations are extremely good. You can learn to manipulate the symbols without mistakes even if you don't really understand the notions. How can you improve on that!