In non-standard analysis the following proof of the product rule is considered valid:
y = uv
Δy = (u + Δu)(v + Δv) - uv
= uΔv + vΔu + ΔuΔv
Δy/Δx = uΔv/Δx + vΔu/Δx + ΔuΔv/Δx
dy/dx = st(uΔv/Δx) + st(vΔu/Δx) + st(ΔuΔv/Δx)
= u dv/dx + v du/dx + 0 * st(Δv/Δx)
= u dv/dx + v du/dx
I would like to understand why st(ΔuΔv/Δx) = 0 without resorting to limits. What I am not understanding is how one can apply the "Increment Theorem" (which states Δy = f'(x) + 𝜀Δx for some infinitesimal 𝜀) without knowing before hand what f'(x) is. How can we even know that ΔuΔv/Δx is infinitesimal without looking to the limit, and even if we knew this how would we know if we had the correct 𝜀Δx?
Is this from this? https://people.math.wisc.edu/~keisler/keislercalc-3-17-21.pdf
I don't understand it either, but I also have a different question. By theorem 3 on page 37, if Δu is infinitesimal, why can't we write
st(vΔu/Δx)=st(v)*st(Δu)*st(1/Δx)=v*0*st(1/Δx)=0
>>1
I've likely got it. Δv/Δx and Δu/Δx have standard-parts dv/dx and du/dx respectively. This implies Δu and Δv/Δx are in the domain of the standard-part function and so we can use its identities. st(ΔuΔv/Δx) = st(Δu) * st(Δv/Δx) = 0. I have no idea why the book mentions the increment theorem in this proof as it is neither necessary nor useful so far as I can tell, but I would love a correction on this.
>>2 aye.
>>3
The standard-part function is undefined for infinite numbers like 1/Δx. An infinitesimal divided by another infinitesimal is indeterminate e.g. 𝛿=𝜀2 such that 𝜀/𝛿 = 1/𝜀. As you don't know if this fraction is in the domain of the standard-part function and you can't use its identities (which were presumably proven for its domain). In this case we know the real part of Δu/Δx so we just evaluate it to that, but for non-zero infinitesimals in general if you find a hole in the graph given the assumption that the infinitesimal is zero you should factor etc. so as to remove this hole or to prove that the standard-part is undefined independent of this.
>>4
Even better would be to claim |ΔuΔv/Δx| < |Δv/Δx| and so the former is within the domain.
>>4
Look at the assumptions of the proof and what the Increment Theorem says. The proof assumes that du/dx, which is the same as u'(x), exists and that Δx is a non-zero infinitesimal. The Increment Theorem says that if u'(x) exists and Δx is infinitesimal, then Δu is infinitesimal (and can be written in a certain form that does not concern us). That's the Δu that the proof uses, and the theorem is mentioned to justify st(Δu)=0. If Δu was not known to be infinitesimal, st(Δu) could be any real number.
>>6
ah okay.
>>6,7 also thank you for taking the time to reason about this it is very heplful to me. I'm just a bit tired is why I wasn't more enthused.
I was meaning to check out non-standard analysis too. Is this book any good?
>>9
Well, it's not an analysis book. Proof theoretical questions are generally put in the "Extra Exercises" section at the end of each chapter. On occasions proofs are also not initially given, or relegated to the appendix as is the case with the standard-part function, and d(e^x)/dx. The foundations of the hyperreal numbers given in the book is weak, and exposition is relegated to another book: "Foundations of Infinitesimal Calculus" which is effectively only the proofs and theorems necessary for (and those which are part of) the book without other content.
What is it? It's a good book on the standard Calculus sequence. It has great breadth covering more than is typically in that sequence, and I think promotes good intuition. The infinitesimals seem to make the proofs given easier to understand, and makes the geometry more solid. To me there are times where things are not sufficiently explained even in terms of just establishing intuition, but I mostly fault myself for not understanding, and eventually come to an understanding.
What is the difference between analysis and calculus? My impression was that they are used interchangeably.
>>11
Sometimes folks say Calculus or Advanced Calculus to refer to Analysis, but I've never heard it the other way around. The difference is mostly rigor; in Analysis you build up a theory of differentiation and integration in a proof theoretical way from an understanding of set theory and the properties of real numbers (or hyper reals apparently) for the sake of proving further theorems or whether certain theorems are applicable. In Calculus you're typically given more or less a recipe book and not expected to fully understand in a rigorous way why everything works, proofs may be given but you're not really expected to write your own, and you won't be assessed on understanding. Really there probably shouldn't be Calculus in this sense at all (perhaps just "Applied Analysis"). Anyway the only reason I said all this is that if Analysis you'll be a bit disappointed by the mentioned book.
>>12
hmm, that last sentence should read: Anyway the only reason I said all this is that if you're looking for the rigor of what is typically called Analysis you'll be disappointed by the mentioned book.
That makes sense, thanks.
I was meaning to check out non-standard analysis too. Is this book any good?
The more I read this book the more I enjoy it. I'm not sure if it's me or the book but I'm making great progress through it. As an example I don't think I really understood the universal quantification of epsilon in the epsilon-delta definition of limits until I thought more seriously about the relationship between it and the infinitesimal definition of limits (and also without loss of generality arbitrarily restricting epsilon above). This is despite having gone threw the motions of many epsilon-delta proofs.
It sound interesting, but a thousand pages? That would take a decade for me to work through.
>>16
Do one day and you get thru it in less than 3 years.
I can math!
Is dy/dx a fraction in non-standard calculs? If you have dy/dx=df/dx+dg/dx, can you multiply it by dx to get dy=df+dg?
Newton or Liebtniz?
https://youtu.be/8yis7GzlXNM?t=41
nietzche
>>20
Archimedes
rhubarb niner
sub sixer