A better notation for a differential?

In summary, the standard notations for differentials in math, such as ##\frac{df}{dx}## and ##\frac{\partial f}{\partial x}##, have been successful and widely used. While there may be some ambiguity, they are generally accepted and understood. Some alternative notations, like ##y'## and ##\ddot x##, have also been used, but the Liebniz notation is considered a significant improvement.
  • #1
MichPod
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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?
 
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  • #2
MichPod said:
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/
 
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  • #3
fresh_42 said:
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.
 
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  • #4
MichPod said:
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:
$$\left. \frac{df(x)}{dx}\right|_{x = x_0}$$
MichPod said:
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:
$$\left.\frac{\partial f(x, y)}{\partial x} \right|_{(x, y) = (x_0, y_0)}$$
MichPod said:
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.
 
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  • #5
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!
 
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1. What is a differential notation?

A differential notation is a mathematical notation used to represent derivatives and differentials in calculus. It is a shorthand way of expressing the relationship between a function and its rate of change.

2. Why do we need a better notation for differentials?

The traditional notation for differentials, using prime symbols and Leibniz's notation, can be confusing and difficult to work with. A better notation can make it easier to understand and manipulate derivatives and differentials in calculus.

3. What are the benefits of a better differential notation?

A better differential notation can make it easier to differentiate between variables and constants, simplify complex expressions, and allow for more efficient calculations. It can also make it easier to understand the relationship between functions and their derivatives.

4. How is a better differential notation different from the traditional notation?

A better differential notation may use different symbols or formatting to represent derivatives and differentials. It may also have specific rules or guidelines for how to use and manipulate the notation.

5. Are there any commonly used notations for differentials?

Yes, there are several commonly used notations for differentials, such as the Newtonian notation, Lagrange's notation, and the dot notation. However, there is ongoing research and discussion about what constitutes a "better" notation for differentials.

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