So, what is the deal with differentials and infinitesimals in physics?

[milesyoung]

I'm currently taking several physics courses (mechanics, thermodynamics etc) and common to them all is their frequent use of infinitesimals.

I'll just give a short recap of how I was taught calculus, and this is how my math teacher would word it:

[calculus training]

$$\frac{dy}{dx}$$ is not a fraction. $$\frac{d}{dx}$$ is a differential operator, y is a function.

The so-called differentials *sneer* can be defined for a function $$f(x)$$ as $$dy=f'(x)\,dx$$.

[/calculus training]

As I was taught calculus I really never heard much about differentials, let alone infinitesimals.

In physics it seems to be a whole other world. Differentials, as in infinitesimal quantities (and not just some watered down form of linear approximation), seem to occur naturally in any mathematical argument, as if the concept of limit etc. weren't really needed.

In the textbook I got for my university calculus course, they go to great lengths to point out that we're not operating with these, apparently distasteful, infinitesimals. We use limits like proper men... and then some pages after, with regards to integrals, infinitesimals are used as a "useful heuristic device for setting up integrals".

It's like "THIS IS WRONG, IT MIGHT BREAK! ... but here, use it anyway".
And they make no effort to tell you when it might break or how it should be used to produce correct results.

In the current physics courses I'm taking they throw differentials around like it's nothing, and it's frustrating to me because I see no logic in it. It's not the way they taught me calculus.

Is there any book out there with the title "This is how you bridge the gap between calculus in math and physics"?

 

[confinement]

The best advice is to always keep in mind what the diffierentials stand for in terms of limits, and to remember that functions in physics are usually uniformly continuous and differentiable and integrable on whatever regions are appropriate. Most of the cases that break the simple logic of infinitesimals are pathological i.e. intentionally ill-behaved cases constructed by Mathematicians to show the shortcomings of working with infinitesimals.

 

[CompuChip]

Actually there's a whole field in between, it's called differential geometry.
There we can actually rigorously define the differential df of a function f. Then if we have some coordinates x, y, ... we can make some basis dx, dy, ... of differentials and df expressed in this basis looks like
$$\mathrm df = \frac{\partial f}{\partial x} \, \mathrm dx + \frac{\partial f}{\partial y} \, \mathrm dy + \cdots$$
where the coefficients are simply the partial derivatives (defined by a limit like you learned in calculus).

Then there's a whole theory of dual bases, differential operators, etc. which gives a well-defined meaning to all those things that physicists tend to write down, like
$$\frac{dx}{dt} = f(t)$$
so
$$dx = f(t) \, dt$$
(for example, when doing a change of variables in an integral).

 

[arildno]

I'm currently taking several physics courses (mechanics, thermodynamics etc) and common to them all is their frequent use of infinitesimals.

I'll just give a short recap of how I was taught calculus, and this is how my math teacher would word it:

[calculus training]

$$\frac{dy}{dx}$$ is not a fraction. $$\frac{d}{dx}$$ is a differential operator, y is a function.

The so-called differentials *sneer* can be defined for a function $$f(x)$$ as $$dy=f'(x)\,dx$$.

[/calculus training]

As I was taught calculus I really never heard much about differentials, let alone infinitesimals.

In physics it seems to be a whole other world. Differentials, as in infinitesimal quantities (and not just some watered down form of linear approximation), seem to occur naturally in any mathematical argument, as if the concept of limit etc. weren't really needed.

In the textbook I got for my university calculus course, they go to great lengths to point out that we're not operating with these, apparently distasteful, infinitesimals. We use limits like proper men... and then some pages after, with regards to integrals, infinitesimals are used as a "useful heuristic device for setting up integrals".

It's like "THIS IS WRONG, IT MIGHT BREAK! ... but here, use it anyway".
And they make no effort to tell you when it might break or how it should be used to produce correct results.

In the current physics courses I'm taking they throw differentials around like it's nothing, and it's frustrating to me because I see no logic in it. It's not the way they taught me calculus.

Is there any book out there with the title "This is how you bridge the gap between calculus in math and physics"?

Hi, miles!

You are absolutely right; dy/dx is NOT a fraction. Nor will it ever be, and the theory of differential forms doesn't assert that, either.


But, and this is a big but:

A quantity like dy/dx can provably ON OCCASION behave just like a fraction does!
And in much of physics, that is, indeed, how you can treat it!
It is done because it is effective and simplifying to a large extent.


However, also here, as in thermodynamics and elsewhere, there will be instances where you cannot regard dy/dx as a fraction, the most famous being the identity written as:
$$\frac{\partial{y}}{\partial{x}}{\frac{\partial{z}}{\partial{y}}\frac{\partial{x}}{\partial{z}}=-1$$

 

[lurflurf]

[calculus training]

$$\frac{dy}{dx}$$ is not a fraction. $$\frac{d}{dx}$$ is a differential operator, y is a function.

The so-called differentials *sneer* can be defined for a function $$f(x)$$ as $$dy=f'(x)\,dx$$.

[/calculus training]

I do not know why you would be sneering, that is entirely rigorous.
dy/dx=(f'(x)dx)/dx=f'(x)
is valid so long as dx!=0

I think the problem is people mean different things when they treat differentials as fractions.

1) differential forms
-differentials are defined in a way that avoids all difficulties
-thus no difficulties are resolved
-creates a divide between definition and intuition
-still helpful
2) nonstandard analysis
-using logic differentials are defined
-definition more in the spirit of intuition
-difficulties remain
3) physics nonsense
-differentials are defined by intuition
-simple cases resolved correctly (flaws in logical foundation)
-difficult cases resolved incorrectly

 

[milesyoung]

Thanks for the replies.

Let me show you an example. This is from MIT OpenCourseWare, 8.01 Physics I: Classical Mechanics, fall 1999, lecture 11 (this Walter Lewin guy seems to be quite popular, even here in Denmark):

(current subject is the Work-Energy Theorem)

$$\begin{align*} &W_{AB}=\int^B_AF\,dx\\ &F=ma=m\frac{dv}{dt}\Rightarrow dx=v\,dt\\ &W_{AB}=\int^B_Am\frac{dv}{dt}v\,dt=\int^{v_B}_{v_A}mv\,dv \end{align*}$$

And while chalking up the integral

$$\int^B_Am\frac{dv}{dt}v\,dt$$

he actually speaks the words "and look what I can do, I can eliminate time" and he chalks over both dt's as if he just did some elementary algebra.

Now I know that this is of course valid. You could probably do this 117 different rigorous ways and get the same result.
What gets me is that they do the exact same thing in my physics courses and they really do work with these objects like it was elementary algebra, and I'm supposed to get something meaningful out of it.

They perform their algebra with infinitesimals and they look at me and go "I did the algebra, you see now how this is clearly valid" but it's just not, to me anyway, not directly from just that simple algebra. Because the algebra dosn't make sense if they don't explain to me what's going on "behind the scenes". The algebra in itself isn't worth anything to me.

It's like I'm missing some rules for this stuff which enables me to do algebra with infinitesimals while still holding true to the way in which I've been taught calculus.

 

[milesyoung]

@lurflurf: I was trying to describe how my elementary calculus teacher would talk about differentials. He would sneer at the term to the point of foaming at the mouth.

 

[arildno]

Well, I can't say I don't agree with you.

Unfortunately, we have reached a "stable equilibrium" in the pedagogics of teaching the mere algebraic manipulation in physics courses, meaning that any deviation from that will be regarded as hopelessly nitpicky, and is therefore quashed..

After all, it is so much quicker to do it sloppily, and we never do it wrong, so why bother with rigour at all?

 

[milesyoung]

That's the thing though. It seems odd to me that you don't get any information on how these clever manipulations, that seem to work exactly lige ordinary algebra, might break - or better yet, when and why they're guaranteed to work.

If they're always valid for continuous functions and whatnot, it would just be really great to see some material on it etc.

 

[arildno]

Well, it isn't too hard to see whether it is truly valid in anyone particular case.

As long as you have an extensive and firm grasp of the proper MATHEMATICAL way of framing it..

So, advice:
Use some of your spare time to read for example vector calculus and other books of applied maths.
Those will usually give you the missing details

 

[csprof2000]

Many physics books simply assume the domain of discussion is well-behaved enough that you can do a little more than you should do in the general case. If you need a clear demonstration of this, see if you can find a copy of the undergraduate level Quantum Mechanics by Griffiths and read the footnotes.

In summary, every other derivation is flawed, except that it actually works for very esoteric reasons. My particular favorite is when he says he will explain why solutions to the TISE must be continuous, and then cleverly avoids the subject later on. Classic.

 

[CompuChip]

In summary, every other derivation is flawed, except that it actually works for very esoteric reasons. My particular favorite is when he says he will explain why solutions to the TISE must be continuous, and then cleverly avoids the subject later on. Classic.

Yep, sounds like your classical: "After the break we will show that X" followed after the break by "As we've shown before the break, X."