1. I recently read of this proof; is it correct?:

Let equal , than , but , thus , therefore .

If the preceding proof is correct than it seems sufficient to say that all infinitisimals are arbitrary entities, where

2.

3. Originally Posted by Ellatha
I recently read of this proof; is it correct?:

Let equal , than , but , thus , therefore .

If the preceding proof is correct than it seems sufficient to say that all infinitisimals are arbitrary entities, where
The proof is correct. I have no idea wha the last sentence is supposed to mean.

4. Thanks for confirming it. The last sentence means is an element of the set of integers. Since infinitesimals aren't finite numbers, than they aren't integers, but is , which is an integer. This than means that infinitesimals don't exist (just like how a limit might appear to be if the difference quotient is applied to it without being first simplified, but in most cases the limit isn't really ).

5. Originally Posted by Ellatha
Thanks for confirming it. The last sentence means is an element of the set of integers. Since infinitesimals aren't finite numbers, than they aren't integers, but is , which is an integer. This than means that infinitesimals don't exist (just like how a limit might appear to be if the difference quotient is applied to it without being first simplified, but in most cases the limit isn't really ).
Infinitesimals are not elements of the real numbers, or any other number system that you are likely to have seen.

There is a way to make the notion of an infitesimal rigorous. Abraham Robinson did it when he developed what is called "Nonstandard Analysis" and it uses something called the "nonstandard real numbers". The development of this theory takes some knowledge of general topology, since it requires the use of something called an untrafilter.

My advice is that, unless you have specific interest in this area and the background to pursue it, that you simply forget about infintesimals. It turns out that they are not generally useful and most (I'm not sure "all" applies) things that you can prove using nonstandard numbers you can also prove using conventional methods. At one time there was some result in the theory of Hilbert space operators that was proved using nonstandard analysis for which no one had found and conventional proof (Paul Halmos was trying) but I don't know if a conventional proof has been found since then (and I can't remember the theorem anyway).

6. Alright. I probably still cannot understand mathematics similar to more advanced analysis (complex analysis, etc.), so I won't read up on the nonstandard analysis you aformentioned. But about forgetting infinitesimals, I was told in the past that the Leibniz concept of the derivative is a ratio of two infinitisemals, namely .

7. Originally Posted by Ellatha
Alright. I probably still cannot understand mathematics similar to more advanced analysis (complex analysis, etc.), so I won't read up on the nonstandard analysis you aformentioned. But about forgetting infinitesimals, I was told in the past that the Leibniz concept of the derivative is a ratio of two infinitisemals, namely .
Maybe, but that is not always a good way to think about it -- as the ratio of two quantities , neither of which are defined.

Sometimes it is useful and sometimes it is just confusing. That is common with notions that are not well-defined.

Basically the derivative is an approximation to a function, locally, by a linear function. That concept works in ordinary calculus of a single variable and generalizes to other situations very well. It extends to the derivative in several variables and a linear operator, and to the notion of the differential of a function in differential geometry, where df is defined without any reference to "infinitesimals".

8. I always did like that definition of a derivative more, that is the slope of the tangent line to some point of a function of , which can be derived using the formula , as in its most simplified form.

Would the "best" [or most general] way of thinking of an indefinite integral as the area under a curve, and the definite integral as the area under the interval of a curve?

9. Originally Posted by Ellatha
I always did like that definition of a derivative more, that is the slope of the tangent line to some point of a function of , which can be derived using the formula , as in its most simplified form.

Would the "best" [or most general] way of thinking of an indefinite integral as the area under a curve, and the definite integral as the area under the interval of a curve?

What generalizes to higher dimensions and is really the most useful way to think of derivatives, at a point is

, as . This reinforces the notion of the derivative as a linear opeator, which is the key to understanding the derivative.

The indefinite integral is nothing more and nothing less than an anti-derivative.

The significant theory involves the definite integral,. The link back to the derivative is the Fundamental Theorem of Calculus which motivates a good deal of the reason for studying indefinite integrals. You don't get to appreciate this until you have studied integration rigorously. That is typically an advanced undergraduate or beginning graduate course. There is more to it than you probably realize.

There is a recent book on the subject (there are lots of books on the subject), based in part on the author's notes from a class once taught by Kakutani at Yale. It strikes me as a very good introduction to general measure and integration. http://www.wiley.com/WileyCDA/WileyT...47025954X.html

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