I’ve always been curious about differentials and how to build a rigorous theory of what the fuck dx, dy, dy/dx, etc. are. For example, if you study Tao’s Analysis and Analysis 2, you will not see anything at all about differentials, and I think that maybe you won’t see the dy/dx notation at all. So, can anyone recommend a textbook about differentials?
Depends what you're looking for. Full Frontal Calculus[0], Intuitive Infinitesimal Calculus[1], and Elementary Calculus[2] are all textbooks on the calculus sequence using an infinitesimal pov. The basic approach is to extend the Real numbers to include infinitesimals (greater than zero but smaller than every positive real number) and transfinites (greater than every positive real number), collectively called the Hyperreals.
If you're looking for a more formal approach, ie the infinitesimal analogue to the usual real analysis, it's called nonstandard analysis and you could probably start with the original, eponymous book written by the creator, Abraham Robinson, for which I unfortunately don't have a link.
If this stuff interests you btw I would also check out Knuth's book on surreal numbers[3], which I believe, in some sense, are the fullest possible extension of what we think of as numbers? But it's been a while since I read into those.
If we're going the hyperreal route, I quite like Goldblatt's GTM Lectures on the Hyperreals. You have to augment it with a paper or two if you want to work with other nonstandard objects, but when I was doing my graduate work it was the resource I kept going back to for clarity.
This looks like the most in-depth resource on the topic that I've seen so far; thanks for adding it! One of the reasons that I'm partial to the hyperreals is because it's such a natural thing, in the context of mathematical history, to extend the number system when that system isn't expressive enough to solve the problems we want to solve. The limit-based approach seems clumsy in comparison.
That same idea applies very nicely to other ideas of math too. Take Konig's Lemma (every infinite, locally finite, connected graph has an infinite path) as an example. The nonstandard proof goes something like:
1. For every natural, you can find a path of that length.
2. Therefore (nonstandard chicanery), for every hypernatural you can find a hyperpath of that hyperlength. Pick one for some infinite hypernatural.
3. Restricting that hyperpath to the original graph yields the infinite normal path you were looking for.
Whole problems melt away entirely as soon as you don't have to worry about clumsy "limit-based" approaches.
Comments
I’ve always been curious about differentials and how to build a rigorous theory of what the fuck dx, dy, dy/dx, etc. are. For example, if you study Tao’s Analysis and Analysis 2, you will not see anything at all about differentials, and I think that maybe you won’t see the dy/dx notation at all. So, can anyone recommend a textbook about differentials?
Depends what you're looking for. Full Frontal Calculus[0], Intuitive Infinitesimal Calculus[1], and Elementary Calculus[2] are all textbooks on the calculus sequence using an infinitesimal pov. The basic approach is to extend the Real numbers to include infinitesimals (greater than zero but smaller than every positive real number) and transfinites (greater than every positive real number), collectively called the Hyperreals.
If you're looking for a more formal approach, ie the infinitesimal analogue to the usual real analysis, it's called nonstandard analysis and you could probably start with the original, eponymous book written by the creator, Abraham Robinson, for which I unfortunately don't have a link.
If this stuff interests you btw I would also check out Knuth's book on surreal numbers[3], which I believe, in some sense, are the fullest possible extension of what we think of as numbers? But it's been a while since I read into those.
[0] https://www.bravernewmath.com/ [1] https://intellectualmathematics.com/calculus/ [2] https://people.math.wisc.edu/~hkeisler/keislercalc-06-03-26.... [3] https://people.math.harvard.edu/~knill/teaching/mathe320_201...
If we're going the hyperreal route, I quite like Goldblatt's GTM Lectures on the Hyperreals. You have to augment it with a paper or two if you want to work with other nonstandard objects, but when I was doing my graduate work it was the resource I kept going back to for clarity.
This looks like the most in-depth resource on the topic that I've seen so far; thanks for adding it! One of the reasons that I'm partial to the hyperreals is because it's such a natural thing, in the context of mathematical history, to extend the number system when that system isn't expressive enough to solve the problems we want to solve. The limit-based approach seems clumsy in comparison.
That same idea applies very nicely to other ideas of math too. Take Konig's Lemma (every infinite, locally finite, connected graph has an infinite path) as an example. The nonstandard proof goes something like:
1. For every natural, you can find a path of that length.
2. Therefore (nonstandard chicanery), for every hypernatural you can find a hyperpath of that hyperlength. Pick one for some infinite hypernatural.
3. Restricting that hyperpath to the original graph yields the infinite normal path you were looking for.
Whole problems melt away entirely as soon as you don't have to worry about clumsy "limit-based" approaches.
Try https://en.wikipedia.org/wiki/Elementary_Calculus:_An_Infini...
Generally most of these 'handwavy' notations are rigidly provable, but only under general assumptions, that might not be true in special cases.
Manfredo P. do Carmo: Differential Forms and Applications. Short and incredibly beautiful.
https://news.ycombinator.com/item?id=49308281