multiplying by dx

jgens

Not quite! In non-standard analysis the derivative of a function is not defined as the quotient of two infinitesimal numbers, but rather their "standard part." Hence, we arrive at the definition:

[tex]\frac{dy}{dx} = \mathrm{st}\left (\frac{f(x + \epsilon) - f(x)}{\epsilon} \right )[/tex]

Note: If [tex]f[/tex] is differentiable, then [tex]f(x + \epsilon) - f(x)[/tex] is infinitesimal if [tex]\epsilon[/tex] is infinitesimal. Also, if [tex]\Delta{y}[/tex] and [tex]\Delta{x}[/tex] are infinitesimal, then their quotient is infinitely close to the function's derivative.

Phrak

To be honest, it was an intentionally provocative claim, and I do appreciate all the criticism, but it really depends on how one defines a fraction--a quotient of reals or a quotient that may include infinitesimals in quotient or denominator, doesn't it?

Hurkyl

The phrase "the derivative is a fraction" is most commonly to mean the assertion
[tex]f'(x) = \frac{f(x+\epsilon) - \f(x)}{\epsilon}[/itex]
for... well... people usually aren't clear on what they really think [itex]\epsilon[/itex] means here.

Of course, one can find all sorts of equations that assert the derivative is equal to a fraction. For example, if

[tex]f(x) = \frac{1}{1 + x^2}[/tex]

then

[tex]f'(x) = \frac{-2x}{(1 + x^2)^2}[/tex]

Voila! The derivative is a fraction!


Even classicaly, I might be inspired to define, for any univariate function f, a binary function (which I will boldly call df) defined by

[tex]d_\epsilon f(x) = \epsilon f'(x)[/tex]

and whose domain is nonzero [itex]\epsilon[/itex] and those x for which f is differentiable

Then, we would have the equation

[tex]\frac{d_\epsilon f(x)}{d_\epsilon x} = f'(x)[/tex]

but this is very different from what the phrase "the derivative is a fraction" is most commonly used to mean.


(Incidentally, the function df I write above is very closely related both to differential forms and to what one might find in a nonstandard analysis textbook such as Keisler's online book)

Phrak

I've read one section by Keisler's calc text, as you recommended. The sections relevant to nonstandard analysis seem to be confined to chapter 1, sections 1.4, 1.5 and 1.6. It's an elementary text, as claimed, so the chapter on vector calculus, unfortunately, doesn't address differential forms.

Phrak

There are plenty of online reference to differential forms but I haven't found any developed, rigorously upon infinitesimals.

Like, what are these things, really?

[tex]\frac{d}{d \lambda } = \frac{ dx^i }{ d \lambda } \frac{ \partial }{\partial x^i} [/tex]

[tex] \nabla = \frac{ \partial }{ \partial x^i } dx^i[/tex]

Hurkyl

The impression I got from a non-standard analyst is "I have infinitessimals -- why would I bother with such things?"

But I did sit down with him and worked out enough of the foundation that it seemed clear to me.

If you fix a "length scale" -- i.e. some nonzero infinitessimal number e, then you can define two relations on points of *Rⁿ:

x ~ y if and only if d(x,y) is a finite multiple of e
x # y if and only if d(x,y) is an infinitessimal multiple of e

The relevant quality of differentiable functions is

if x~y, then f(x)~f(y)
if x#y, then f(x)#f(y)

The tangent space to any standard point P is simply the set of all points such that P~x, modulo the equivalence relation #. (i.e. two points x and y name the same tangent vector if and only if x#y)

It's not difficult to show this is a vector space -- e.g. that the sum of two tangent vectors is the same (modulo #) no matter what differentiable coordinate chart you use to compute it.

Phrak

I'm sorry I can't follow that. What are the symbols # and ~ ?

Hurkyl

Binary relations whose definition is given in the first indented section of my post.

Phrak

You're way ahead of me. I'd need a short course in differential forms presented mathematically, rather than for relativity, to fully appreciate your post.