Abuse of Notations: Why Is dy/dx Abused?

[xphysics]

Hey guys, umm I'm just going to get straight to the point, why is the notation of dy/dx for example is being abused so badly? it's not a fraction right? because you can express it in Newtonian notation like y' too! i mean it's just a notation to represent the change in y respect to x
I was learning about the capstan equation for example and derivation abused it like this:
http://en.wikipedia.org/wiki/Capstan_equation
look at (7), how could you express d/dx= ?
can someone explain this to me? please

 

[Simon Bridge]

Hey guys, umm I'm just going to get straight to the point, why is the notation of dy/dx for example is being abused so badly?

Because to do so is useful. How do you mean "abuse"? How is it abusive to utilize a standard and well accepted form of notation?

it's not a fraction right?

Can you tell us why it is not a fraction - how are it's properties different from the properties of a fraction?

i mean it's just a notation to represent the change in y respect to x

I don't think so... the notation to represent the change in y wrt x is actually $$\frac{\Delta y}{\Delta x}$$ ...

look at (7), how could you express d/dx= ?

How would you express the same idea?

can someone explain this to me? please

The equivalent in "change of" notation is $$\frac{\Delta}{\Delta x}$$... it is a construction called an operator.
You seemed perfectly happy with dy/dx as an abstract notation and now you have a problem with something that is abstract notation?
http://en.wikipedia.org/wiki/Leibniz's_notation

 

[xphysics]

but...it's not a fraction?

 

[rtsswmdktbmhw]

Change in y with respect to x is a fraction Δy/Δx

dx and dy respectively represent infinitesimally small x and the y value of that infinitesimally small x, so dy/dx is a fraction as well (it's the same as Δy/Δx with Δx→0 but written as dy/dx instead). That's how I think of it.

 

[dauto]

This is not an abuse of notation. An abuse of notation is a non-standard potentially confusing notation. That's not what we have here. dy is the standard notation for a differential, dy/dx is the standard notation for a derivative, and d/dx is the standard notation for a differential operator.

 

[Simon Bridge]

but...it's not a fraction?

... prove it!

Your own example shows that it shares many of the same properties as a fraction - so what's the difference?

Even if we accept that it is not a fraction - so what?
Does it have to be a fraction?
Can it not be some other entity that shares many of the same properties as a fraction?
Why does it matter?

You have not shown any examples of "abuse" of the notation - only it's normal accepted use.
The only objection seems to be that you don't like it.

In other words: what is the problem?

 

[micromass]

This is a funny thread. You would have gotten completely different answers in the mathematics section of the forum. Of course $\frac{dy}{dx}$ is not a fraction formally. I don't know how you could claim that it is. But physicists like to abuse the notation and see it as a fraction anyway. It's a very useful abuse of notation. The abuse of notation can be explained formally by nonstandard analysis which allows infinitesimals.

 

[xphysics]

Okay, yes a person that agrees with me that dy/dx isn't a formal "fraction" i mean the literal translation of it is "change in y divided by change in x" but that's not it, it's just "the change in y with respect to x." because we are calculating the change infinitesimally. i heard my teacher said that real analysis actually justifies the expression in fraction for the abuse.

 

[micromass]

Okay, yes a person that agrees with me that dy/dx isn't a formal "fraction" i mean the literal translation of it is "change in y divided by change in x" but that's not it, it's just "the change in y with respect to x." because we are calculating the change infinitesimally. i heard my teacher said that real analysis actually justifies the expression in fraction for the abuse.

You are correct. Except that it's not real analysis that justifies the situation, but nonstandard analysis.

 

[Simon Bridge]

For a given definition of "non-standard"... also a given definition of "abuse".
I think xphysics has been told it is an abuse of the notation but does not understand why it is caled abuse.
Perhaps you'd like to explain? :)

Note: In common language, "abuse" implies there is something incorrect or immoral about it.
Don't let this abuse continue!1!

 

[micromass]

For a given definition of "non-standard"... also a given definition of "abuse".
I think xphysics has been told it is an abuse of the notation but does not understand why it is caled abuse.
Perhaps you'd like to explain? :)

Note: In common language, "abuse" implies there is something incorrect or immoral about it.
Don't let this abuse continue!1!

I think xphysics understands perfectly well why it's an abuse. It's because $\frac{dy}{dx}$ is not a fraction because it does not equal $dy$ divided by $dx$. It does not equal that because there is no definition of what $dy$ and $dx$ is. We can define what it is using differential geometry, but that avoids the issue. Using nonstandard analysis is the only real approach that works because then you can actually rigorously define infinitesimals.

 

[Simon Bridge]

Hopefully that is it - then the question is now completely answered.

 

[Matterwave]

I think xphysics understands perfectly well why it's an abuse. It's because $\frac{dy}{dx}$ is not a fraction because it does not equal $dy$ divided by $dx$. It does not equal that because there is no definition of what $dy$ and $dx$ is. We can define what it is using differential geometry, but that avoids the issue. Using nonstandard analysis is the only real approach that works because then you can actually rigorously define infinitesimals.

Would a mathematician prefer $\frac{d}{dx}(y)$? Or $y'(x)$? But in the latter notation, one won't know what to take the derivative with respect to if $y$ is a function of many variables.

I would never call $\frac{dy}{dx}$ an abuse of notation simply because everyone knows exactly what it means. I would certainly call $dy\div dx$ an abuse of notation (oh my eyes!), but not $\frac{dy}{dx}$...

 

[micromass]

Would a mathematician prefer $\frac{d}{dx}(y)$? Or $y'(x)$? But in the latter notation, one won't know what to take the derivative with respect to if $y$ is a function of many variables.

I would never call $\frac{dy}{dx}$ an abuse of notation simply because everyone knows exactly what it means. I would certainly call $dy\div dx$ an abuse of notation (oh my eyes!), but not $\frac{dy}{dx}$...

There is no problem with the $\frac{dy}{dx}$ notation if you keep in mind that it's not a fraction. The abuse I'm talking about happens when you actually do treat it like a fraction. For example when you do something like $\frac{dy}{dx} = 5$ implies $dy - 5dx = 0$. This can be very handy sometimes and I do it too, but it's still abuse.

 

[adjacent]

This can be very handy sometimes and I do it too, but it's still abuse.

Then why don't mathematicians find another way to make it less abused?

 

[Philip Wood]

One could argue that the reason why the Leibniz notation, $\frac{dy}{dt}$ became more popular than Newton's $\dot{y}$ notation was because $\frac{dy}{dt}$ is open to such fruitful abuse...

 

[micromass]

Then why don't mathematicians find another way to make it less abused?

One could argue that the reason why the Leibniz notation, $\frac{dy}{dt}$ became more popular than Newton's $\dot{y}$ notation was because $\frac{dy}{dt}$ is open to such fruitful abuse...

Right. The notation is popular because it can be abused and because the abuse is very useful.
There is nothing wrong with abusing a notation if you know it's abuse and if you know how to do it the rigorous way. Professional physicists and mathematicians know this. The problem is that students and some teachers don't know it and say things like $\frac{dy}{dx}$ IS a fraction. So I would say that Leibniz notation is not a very good pedagogical notation. But if you know it's abuse and if you know how to do it the right way, then there's nothing wrong with abusing a notation.

Finding the right notations is a very important part of mathematics, it can make math easier and harder. You really can't do math without abusing notations now and then. For example, even saying things like $\mathbb{R}\subseteq \mathbb{C}$ is already very dubious. However, doing things the "right" way would make mathematics a lot uglier and less intuitive, so that's not an option. We will just have to live with the abuse, since it makes it a lot more intuitive for us. As long as we realize it's abuse, then it's ok.

 

[dauto]

Right. The notation is popular because it can be abused and because the abuse is very useful.
There is nothing wrong with abusing a notation if you know it's abuse and if you know how to do it the rigorous way. Professional physicists and mathematicians know this. The problem is that students and some teachers don't know it and say things like $\frac{dy}{dx}$ IS a fraction. So I would say that Leibniz notation is not a very good pedagogical notation. But if you know it's abuse and if you know how to do it the right way, then there's nothing wrong with abusing a notation.

Finding the right notations is a very important part of mathematics, it can make math easier and harder. You really can't do math without abusing notations now and then. For example, even saying things like $\mathbb{R}\subseteq \mathbb{C}$ is already very dubious. However, doing things the "right" way would make mathematics a lot uglier and less intuitive, so that's not an option. We will just have to live with the abuse, since it makes it a lot more intuitive for us. As long as we realize it's abuse, then it's ok.

I think your definition of abuse is different from my definition of abuse. If something is a standard notation, and a useful notation, and its use leads to correct answers, and it is understood by everybody, how can it be considered an abuse? dy - 5 dx = 0 isn't an abuse of notation since dy and dx are the standard notation for differentials. Differentials are very useful for doing things such as implicit differentiation.

 

[micromass]

I think your definition of abuse is different from my definition of abuse. If something is a standard notation, and a useful notation, and its use leads to correct answers, and it is understood by everybody, how can it be considered an abuse? dy - 5 dx = 0 isn't an abuse of notation since dy and dx are the standard notation for differentials. Differentials are very useful for doing things such as implicit differentiation.

And how do you define $dy$ and $dx$?

 

[dauto]

And how do you define $dy$ and $dx$?

The exact definition depends on context and desired level of rigor. For most purposes the concept of an infinitesimal is just fine. If you don't like that, there are many approaches that could be used to develop more rigorous definitions but they all get quite complex quite fast and are really not necessary for most applications. The non-standard analysis based on hyperreal numbers was mentioned earlier in this thread by you - if I'm not mistaken. It is one possible way, but not the only one.

 

[micromass]

The exact definition depends on context and desired level of rigor. For most purposes the concept of an infinitesimal is just fine. If you don't like that, there are many approaches that could be used to develop more rigorous definitions but they all get quite complex quite fast and are really not necessary for most applications. The non-standard analysis based on hyperreal numbers was mentioned earlier in this thread by you - if I'm not mistaken. It is one possible way, but not the only one.

Right. And it is my opinion that as long as somebody doesn't know how to rigorously define $dy$ and $dx$, but uses it "as it were numbers in a fraction", then he is abusing notation.

To be clear, I don't have anything against the notation $\frac{dy}{dx}$, but I do have something against how it's taught. If a teacher defines

$$\frac{dy}{dx} = \lim_{h\rightarrow 0} \frac{y(x+h) - y(x)}{h}$$

and then says that the chain rule can be proven because it's just elimination like in fractions, then that's an abuse of notation.

And again, I have nothing against abuse of notation. It is very handy and I do it myself. But you should know it is abuse of notation. This is not always made clear.

 

[Simon Bridge]

It's like any language - there are grammarians who will frown at every "incorrect" structure in a sentence as an "abuse" of the language. Some will oppose any use of slang (say) while others will agree that the slang is a "useful abuse" so long as you know you are misusing the language and are careful about it you are fine.

Also see:
https://www.physicsforums.com/showthread.php?t=318484
http://en.wikipedia.org/wiki/Abuse_of_notation
... for other discussions.

 

[D H]

but...it's not a fraction?

No, it's not. Thinking it is a fraction will get you in trouble with higher order derivatives and with partial derivatives.

Higher order derivatives: Thinking that $\frac {dy}{dx}$ is a fraction might lead you to think that
$$\frac {d^2 f(y(x))}{dx^2} = \frac {d^2 f}{dy^2} \left(\frac {dy}{dx}\right)^2$$ The correct result is
$$\frac {d^2 f(y(x))}{dx^2} = \frac {d^2 f}{dy^2} \left(\frac {dy}{dx}\right)^2 + \frac {d^2 y}{dx^2} \frac {d f}{dy}$$ Treating $\frac {d^2 f(y(x))}{dx^2}$ as a fraction erroneously omits the second term.

Partial derivatives: Consider the unit sphere, $x^2+y^2+z^2 = 1$. Thinking that $\frac {\partial y}{\partial x}$ is a fraction might lead you to think the following for points on the sphere:
$$\frac {\partial x}{\partial z} \frac {\partial z}{\partial y} \frac {\partial y}{\partial x} = 1$$The correct result is the rather non-intuitive
$$\frac {\partial x}{\partial z} \frac {\partial z}{\partial y} \frac {\partial y}{\partial x} = -1$$