How do you Treat Infinitesimals?

[TheDemx27]

All throughout calculus texts, the authors have always put conditions on the manipulation of differentials. They say that for the chain rule, the cancellation of the differentials is simply a way to remember the formula. When doing separation of variable for ODEs, texts always say something like: "Oh, so we group the $dy$'s and $dx$'s on both sides, even though that is kind of a handwavy way of handling the differentials and it's not how they actually work" or something like that. There is always some little side note that the manipulation isn't rigorous. In all my experiences, treating the differentials as variables and manipulating them as such works out fine.

How are you actually supposed to treat infinitesimals then?

 

[andrewkirk]

It is a valuable exercise to work to understand the rigorous justification for the cases where dx terms are treated like ordinary numbers. Usually it's not too hard to understand. For instance the proof of the validity of the chain rule has nothing to do with cancelling out infinitesimals. All cancelling is done on finite quantities inside the limit parentheses. Separation of variables to get from $\frac{dy}{dx}=\frac{f(x)}{g(y)}$ to $\int g(y)dy=\int f(x)dx$ can be done by multiplying both sides by $g(y)$, integrating wrt x and then using substitution and the chain rule.

When you've done that for the main cases, you can apply the rule of thumb that there's a good chance it's illegal if you can't think of a rigorous justification.

However, it's helpful to distinguish between exploration and proof. When one is exploring possible avenues for a proof, it can be efficient to just assume without proof one can manipulate the differentials, in order to go on and see what happens next. If it turns out to be a dead end anyway, one will have saved the time it would have taken to justify the manipulation. If it turns out to not be a dead end, one can then go back and try to justify the manipulation.

 

[HallsofIvy]

Actually, most Calculus texts, all those except the relatively new "non-standard Analysis" texts, don't mention "infinitesmals" at all, they deal with limits instead

 

[TheDemx27]

Separation of variables to get from $\frac{dy}{dx}=\frac{f(x)}{g(y)}$ to $\int g(y)dy=\int f(x)dx$ can be done by multiplying both sides by $g(y)$, integrating wrt x and then using substitution and the chain rule.

$$\frac{dy}{dx}=\frac{f(x)}{g(y)}$$
$$\int g(y)\frac{dy}{dx} dx =\int f(x)dx$$

...?​

I don't know what bit of meaningful substitution you could do here.

 

[andrewkirk]

I don't know what bit of meaningful substitution you could do here.

We use the substitution rule.

 

[TheDemx27]

We use the substitution rule.

$u=y, du=\frac{dy}{dx}dx$
$\int g(y)\frac{dy}{dx} dx=\int f(x)dx$
$\int g(u)du =\int f(x)dx$
and $u=y$ so
$\int g(y)dy=\int f(x)dx$
Is that what you meant?

 

[HallsofIvy]

You confused me by using the word "infinitesmals" in the title of the thread when the thread is really about "differentials"!

 

[andrewkirk]

$u=y, du=\frac{dy}{dx}dx$
$\int g(y)\frac{dy}{dx} dx=\int f(x)dx$
$\int g(u)du =\int f(x)dx$
and $u=y$ so
$\int g(y)dy=\int f(x)dx$
Is that what you meant?

Yes, that's it.

 

[TheDemx27]

You confused me by using the word "infinitesmals" in the title of the thread when the thread is really about "differentials"!

Sorry about that. I guess I don't know the difference between the two. Wikipedia seems to think they are the same thing. I'm guessing differentials are a subset of infinitesimals?

 

[HallsofIvy]

No, "infinitesimals" don't exist at all in the usual real number system. In "non-standard analysis", that I referred to above, "infinitesimals" are developed by constructing an entirely new number system that includes numbers, in addition to the usual real numbers, called "infinitesimals": https://en.wikipedia.org/wiki/Non-standard_analysis

Differentials, on the other hand, are an operation on functions- given any function, f, such that $\frac{df}{dx}= g$, the "differential" is defined as $df= g(x)dx$ and "dx" is essentially left undefined. In ordinary Calculus, "df= g(x)dx" can be approximated by using a small value for dx. In non-standard analysis "differentials" can be identified with "infinitesimals".