Notion of the infinitesimal analysis in R^n

[samspotting]

When I was in calculus, the notion of the infinitesimal, the smallest possible unit, was really emphasized.

When I switched into a more theoretical section of calculus (analysis in R^n), nothing about infinitesimals is said, instead all the proofs are in the style of epsilons and deltas.

Is the infinitesimal just a broad statement about something like for all epsilon strictly greater than zero this must work?

 

[slider142]

The infinitesimal in lower calculus courses is just a way of pushing rigorous limits and limiting behavior under the carpet so that they can deal with just the results of analysis instead of worrying about proving some rather delicate/subtle concepts. However, there is a formulation of analysis that uses the concept of infinitesimal rigorously, this is usually referred to as non-standard analysis.

 

[qntty]

Yeah, the epsilon/delta approach was created because infinitesimals weren't rigorous enough. However, you can still read about them in Keisler's book.

 

[confinement]

Yeah, the epsilon/delta approach was created because infinitesimals weren't rigorous enough.

I would rather say that it was because infinitesimals are too difficult to make rigorous. Less well known than non-standard analysis which appeared in the 1960s is the more recent analysis in smooth worlds:

http://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis

From a mathematicians viewpoint, using infinitesimals to do calculus is like using a howitzer to defeat a bear. They would much rather show that it is possible to defeat the bear with a simple knife i.e. for the development of calculus it will suffice to speak of only real numbers e.g. epsilon-delta formulation.

In intermediate/advanced level physics infinitesimals are used all the time in place of time consuming repetitive arguments dealing with limits.

 

[Hurkyl]

Infinitessimals of the type in non-standard analysis (NSA) are very tricky. They only really work under two circumstances:
1. You pay close attention to the technical details governing their use
2. You avoid doing anything creative, and only regurgitate arguments you've been told are valid

The emphasis in an elementary calculus class is how to make effective calculations, and is completely independent of the theoretical foundation. NSA doesn't change how you integrate by parts. Infinitessimals aren't used when you need to estimate how much error you introduce by using $\sqrt{101} \approx \sqrt{100}$. Infinitessimals aren't used when figuring out how many terms of a Taylor series you need to compute $\sin 1$ to 3 decimal places.

Furthermore, even for the theoretical foundations, the elementary calculus student won't see a practical difference using infintiessimals over epsilon-deltas; e.g. in proving a particular limit formula, you're usually going to do pretty much exactly the same work with inequalities whether you're using epsilon-deltas or you're using infinitessimals.



And people seem to forget that standard techniques do offer infinitessimals -- big-Oh notation, dual numbers, tangent spaces, ...

Generally speaking, (I believe) physicists' usage of infinitessimals fall into two categories:
1. Arguments that pretty much directly translate into perfectly rigorous arguments
2. Arguments that brush away dangerous technical issues

(An example of the latter is thinking of the Dirac delta as being zero everywhere except at the origin, which makes it somewhat difficult to fathom just how badly behaved ideas like $\delta(0)$ or $\delta(x) \delta(x)$ really are)



(For the record, I really do like NSA)

 

[Ben Niehoff]

I think teaching calculus without using epsilon-delta limits is silly. Infinitesimals are more difficult to make rigorous, and are therefore more confusing. Students will be left thinking you are doing voodoo rather than legitimate math. They will also be left with silly notions such that "0.999... != 1", etc., thinking that such expressions can "differ by an infinitesimal".

However, once you understand them as a shortcut for limits, infinitesimals are useful. Essentially, we can define an infinitesimal epsilon as some number such that

$$\epsilon \neq 0$$

and

$$\epsilon^2 = 0$$

This sounds like nonsense if epsilon is taken to be a real number. However, one may consistently define epsilon as a matrix, such as

$$\epsilon = \left( \begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix} \right)$$

Then an ordinary real number would be a diagonal matrix. Sums would work out as expected:

$$1 + \epsilon = \left( \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right) + \left( \begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix} \right) = \left( \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix} \right)$$

$$(1 + \epsilon)^2 = 1 + 2\epsilon + \epsilon^2 = 1 + 2\epsilon$$

$$\left( \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix} \right) \left( \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix} \right) = \left( \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix} \right) = \left( \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right) + 2 \left( \begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix} \right)$$

Then, we can define the derivative as

$$f'(x) = \frac{f(x + \epsilon) - f(x)}{\epsilon}$$

However, actually evaluating such derivatives (without using limits) requires us to already know about Taylor series, if our function is something more complicated such as $\sin x$.

 

[jostpuur]

That presentation is not 100% right, because

$$\frac{1}{\epsilon} = \left(\begin{array}{cc}<br /> 0 & 1 \\<br /> 0 & 0 \\<br /> \end{array}\right)^{-1}$$

does not exist.