Leibniz's Operators: True or False?

• qxcdz
In summary, Leibniz notation in calculus involves using \mathrm d as an operator to create infinitesimals from variables and \int as a special summation operator. However, the concept of infinitesimals was never properly defined, leading to the use of limits instead. Non-standard analysis has attempted to extend the real number system to include infinitesimals, but it requires a deeper mathematical background and does not change the way calculus is taught. The term "differential" is used in calculus to represent a differential operator, which is different from the concept of "infinitesimal". It is a legitimate operation to multiply both sides of an equation by dx, but it is not the same as multiplying by an infinitesimal. In non-standard
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?

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.

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:

$$y=x^2$$
$$y+\mathrm dy = (x+\mathrm dx)^2 \\$$
$$y+\mathrm dy = x^2+2x\mathrm dx+\mathrm dx^2 \\$$
$$\mathrm dy = 2x\mathrm dx+\mathrm dx^2 \\$$
$$\frac{\mathrm dy}{\mathrm dx} = 2x+\mathrm dx \\$$
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?

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.

HallsofIvy said:
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:

$$y=x^2$$
$$y+\mathrm dy = (x+\mathrm dx)^2 \\$$
$$y+\mathrm dy = x^2+2x\mathrm dx+\mathrm dx^2 \\$$
$$y+\mathrm dy = x^2+2x\mathrm dx+0 \\$$
$$y+\mathrm dy = x^2+2x\mathrm dx \\$$
$$\mathrm dy = 2x\mathrm dx \\$$
$$\frac{\mathrm dy}{\mathrm dx} = 2x \\$$

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 ex, 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?

1. What are Leibniz's Operators?

Leibniz's Operators refer to a set of mathematical notations developed by the German mathematician and philosopher Gottfried Wilhelm Leibniz. These notations are used to represent differentiation and integration in calculus.

2. What is the significance of Leibniz's Operators?

Leibniz's Operators revolutionized the field of calculus by providing a more intuitive and efficient way to represent differentiation and integration. They are still widely used today in mathematical and scientific research.

3. Are Leibniz's Operators always true?

Leibniz's Operators are not statements that can be labeled as true or false. They are not equations or theorems, but rather notations used to represent mathematical operations. Whether they accurately represent the underlying mathematical concepts is a matter of mathematical convention and interpretation.

4. Can Leibniz's Operators be used in all types of calculus problems?

Yes, Leibniz's Operators can be used in all types of calculus problems as long as they involve differentiation and integration. They are particularly useful in problems involving multiple variables and complex functions.

5. How can I learn to use Leibniz's Operators?

To learn how to use Leibniz's Operators, it is important to have a solid understanding of calculus and its underlying principles. There are many online resources and textbooks available that can help you learn and practice using these notations.

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