Leibniz's Operators: True or False?

[qxcdz]

I heard something about the well known Leibniz notation of calculus, and I thought that you guys would be able to tell me if it's a load of hogwash or not.

The geist of it is this: \mathrm d and \int are actually operators, with \mathrm d being an operator that creates an infinitesimal from a variable, and \int being a special kind of summation operator. So, whereas now, we'd recognise \int \mathrm dx as being the same as \int 1 \mathrm dx, Leibniz would have seen it as applying an infinitesimal operation to a variable, and then it's inverse.

So, when Leibniz wrote things like \frac{\mathrm dy}{\mathrm dx}=\frac{\mathrm dy}{\mathrm dt}\cdot \frac{\mathrm dt}{\mathrm dx}, he saw it as literally cancelling down fractions, not as a trick with limits that just resembles cancelling down fractions.

Is this really what the notation meant? Is it still valid?

 

[HallsofIvy]

The difficulty is that Leibniz was never able to define "infinitesmal" properly. The Bishop Berkeley famously satirised them as "ghosts of departed quantities". That's why the concept of "limits" was used to develop calculus instead.

In the last half century, a lot of work has been done in developing "non-standard analysis" which starts by extending the real number system to include "infinitesmals" (and their reciprocals, "infinite numbers"). In order to do that, you have to go back to basic concepts of logic (in particular the "compactness property": if every subset of a collection of axioms has a model, then the entire set has a model). Since the results are exactly the same as in standard calculus, but requires a lot deeper background to be rigorous, it's not going to change the way calculus is taught.

 

[qxcdz]

So, infinitesimals are not normally used in calculus, really? I know that \frac{\mathrm dy}{\mathrm dx} is a function, and not a ratio, but my maths teacher has called \mathrm dx an 'infinitesimal' quite a few times. He's also multiplied both sides of an equation by it, and then apparently justified it despite \frac{\mathrm dy}{\mathrm dx} not being a fraction, by putting integral signs on both sides of the equation.

I'm going to guess that was all mathemagic, that, like the chain rule, just happens to look exactly like treating \frac{\mathrm dy}{\mathrm dx} as a fraction.

About that nonstandard analysis bit, it sounds interesting, and I accept that giving a full explanation would probably need several years of background I don't have, but:

<br /> y=x^2<br />
<br /> y+\mathrm dy = (x+\mathrm dx)^2 \\<br />
<br /> y+\mathrm dy = x^2+2x\mathrm dx+\mathrm dx^2 \\<br />
<br /> \mathrm dy = 2x\mathrm dx+\mathrm dx^2 \\<br />
<br /> \frac{\mathrm dy}{\mathrm dx} = 2x+\mathrm dx \\<br />
Replace all the d with deltas, and add a limit at the end, and that's how I saw differentiation defined. However, there seems to be the problem of an infinitesimal quantity 'left over' at the end, that should not be there. Are you allowed to simply discard it, as Leibniz did, or is there some mathematical trickery that has to be pulled?

 

[HallsofIvy]

Yes, "infinitesmal" is a nice concept as long as you don't have to define it rigorously! If you really want to annoy your teacher, ask him/her to define "infinitesmal"! Of course, your grade might suffer. It is, in fact, perfectly legitimate to "multiply both sides of an equation by dx" but that's multiplying by the differential, not an "infinitesmal". There's a very large difference in the way a "differential" is defined and the way "infinitesmal" is defined. "Differential" can be (and is) defined in any Calculus book. To define "infinitesmal", you need to expand the real number system in a very abstract way.

 

[qxcdz]

Yes, "infinitesmal" is a nice concept as long as you don't have to define it rigorously! If you really want to annoy your teacher, ask him/her to define "infinitesmal"! Of course, your grade might suffer. It is, in fact, perfectly legitimate to "multiply both sides of an equation by dx" but that's multiplying by the differential, not an "infinitesmal". There's a very large difference in the way a "differential" is defined and the way "infinitesmal" is defined. "Differential" can be (and is) defined in any Calculus book. To define "infinitesmal", you need to expand the real number system in a very abstract way.

Well, looking through my sources, i.e. google, I found out that infinitesimals, as usually defined in "non-standard analysis", are "nilpotent", so, it I've understood this right, it changes the derivation to:

<br /> y=x^2<br />
<br /> y+\mathrm dy = (x+\mathrm dx)^2 \\<br />
<br /> y+\mathrm dy = x^2+2x\mathrm dx+\mathrm dx^2 \\<br />
<br /> y+\mathrm dy = x^2+2x\mathrm dx+0 \\<br />
<br /> y+\mathrm dy = x^2+2x\mathrm dx \\<br />
<br /> \mathrm dy = 2x\mathrm dx \\<br />
<br /> \frac{\mathrm dy}{\mathrm dx} = 2x \\<br />

Which works. Not that I really understand how it makes sense to say that \mathrm dx^2=0.

But, I drew a blank on "differential" google gives me pages about the internal mechanisms of a car. I even looked through my textbooks. We don't have textbooks specifically for calculus over here in the Land of the Lime-Eating Sailors, but I do have Core Mathematics 1, 2 and 3. None of which contain a definition of the term differential. And these are supposed to be beyond 'high-school' level, C3 contains the chain, product, and quotient rules, as well as e^x, ln x, etc.

I guess, judging by my luck, that the definition is in C4, which although we have done work from, we don't have 'take-home' copies of yet. That's where multiplying by dx is introduced. So, please, could you define 'differential' for me?