Why does MTW keep calling the "product rule" the "chain rule"?

[FreeThinking]

TL;DR

MTW page 257 exercises 10.2 through 10.5 calling product rule as chain rule.

MTW p 257, exercises 10.2 through 10.5: These exercises are all dealing with this familiar property of derivatives ∇ (AB) = ∇A B + A ∇ B . I learned this was called the "product rule". I learned that d/dx f(y(x)) = df/dy dy/dx is called the "chain rule". MTW keeps calling what I learned as the "product rule" by the term "chain rule". I've Googled and such and all the hits use the terms in the way I expect. Why is MTW calling them differently? What am I not understanding?

Also, they do this in other places in the book also.

Thanks.

 

[fresh_42]

My guess is, that it is either a spleen or a leftover from differentials as boundary operators of (co-)chain complexes, a rule how to deal with chains or cochains so to say.

The product rule is also called Leibniz rule or Jacobi identity or boundary operator. Chain rule is new to me.

 

[strangerep]

MTW p 257, exercises 10.2 through 10.5: These exercises are all dealing with this familiar property of derivatives ∇ (AB) = ∇A B + A ∇ B . I learned this was called the "product rule". [...]

Perhaps because different authors wrote different parts?

E.g., near the top of p182, and near the top of p216, they do call it the "product rule". But on p76, near eq(3.21), the use the name "product rule" for writing tensor products in component form.

 

[FreeThinking]

@fresh_42: Thank you for response. Unfortunately, I have no idea what you said. I looked up the word "spleen" but could find no definition that fit the current context. And I have no idea what "(co-)chain complexes" are. Whatever they are, I hope that's not what MTW meant or I might as well give up any attempt to understand this subject. :-(

Calling the "product rule" the "Leibniz rule" is also in keeping with my understanding, although I can't remember it being called the "Jacobi identity" or "boundary operator". I will have to research those. Thanks for that info.If the chain rule is new to you, then I didn't state it well in my post, so let me try again. The chain rule I'm talking about is from elementary calculus and refers to how you take the derivative of a function composition, namely:

Let y=g(x) be a continuous function of x, and let z = f(y) be a continuous function of y. Then the derivative of z with respect to x is just ${\frac {dz} {dx}} = \frac {dz} {dy} \frac {dy} {dx}$. At least that's my understanding of the Google hits I get on it. The above equation appears nowhere in the exercises or other places where they say "chain rule".

My problem is that what MTW calls the "chain rule" looks more like the product rule to me: d(fg)=df g + f dg . All my hits on Google seem to confirm my view. That is the equation that appears in all those places where they say "chain rule".

Since MTW are 3 of the best physics/mathematics minds of our age, I figured the failure in understanding is most certainly mine. I just didn't know where to go or who to ask, since I can find nothing explaining it on the internet.

Thanks.

 

[PeterDonis]

Since MTW are 3 of the best physics/mathematics minds of our age, I figured the failure in understanding is most certainly mine.

What you're failing to understand isn't anything of substance: you clearly understand how to take derivatives of products of two functions and of function compositions. Which of those you use the terms "product rule" or "chain rule" to refer to is a matter of terminology, not math or physics. I don't think anyone except MTW themselves can answer the question of why MTW chose particular terminology in a particular place in the book. I don't understand why they picked that particular terminology either. But I wouldn't spend much time worrying about that.

 

[fresh_42]

Thank you for response. Unfortunately, I have no idea what you said. I looked up the word "spleen" but could find no definition that fit the current context. And I have no idea what "(co-)chain complexes" are. Whatever they are, I hope that's not what MTW meant or I might as well give up any attempt to understand this subject. :-(

Maybe I have chosen a wrong word. We use it in my language and as it sounded and is written English I thought it would exist. I meant quirk or personal peculiarity.

Chain complexes are described in Wikipedia. The boundary operator of the Chevalley Eilenberg complex for Lie algebras follow the same rules as differential operators aka covariant derivatives on n-forms do. They are closely related. This would have been a possibility for the wording, but as I said, a guess.

If the chain rule is new to you, then I didn't state it well in my post, so let me try again.

It is new to call the Leibniz rule chain rule. The Jacobi identity is the product rule for vector fields.

 

[FreeThinking]

@strangerep: Thanks for the additional references. I'll check them out.

I can believe different authors wrote different parts. In fact, I would suspect that more than a few graduate students got thrown into the mix as well.

My problem is that MTW is my first exposure to a lot of the math with which I'm struggling so I don't know enough to spot the problems if they're there. I have compared many of their important equations to other sources and have found no discrepancies that I can detect. That has lead me to trust them.

So, do you think it's safe to assume they were that sloppy with their nomenclature and I can just quit worrying about it? If so, I find that prospect very distressing.

Thanks.

 

[PeterDonis]

do you think it's safe to assume they were that sloppy with their nomenclature and I can just quit worrying about it?

I think you can trust the equations in MTW, and those are what contain the substantive content. If the ordinary language discussion uses a term that doesn't seem appropriate to you for a particular equation, you can just ignore the term and look at the equation. Nothing of substance will be affected.

If so, I find that prospect very distressing.

Why? Why does the ordinary language matter if you have the equations?

 

[PeterDonis]

MTW is my first exposure to a lot of the math

MTW is probably not the best first textbook on GR to learn from. It is certainly comprehensive, but for that very reason it contains a lot of material that really isn't necessary if you are just trying to learn the basics of GR.

If you haven't looked at Sean Carroll's lecture notes on GR, you might give them a try:

https://arxiv.org/abs/gr-qc/9712019
They cover the basics, including the basics of the math--manifolds, tensors, differential geometry--in a much more focused way that might be easier as an introduction.

 

[FreeThinking]

@fresh_42: Ok, now I understand you. I've never known the word "spleen" to mean that, but apparently it is used, at least in the U.S, I don't know about anywhere else, to mean "complain" or "rant". So, thanks for your guess. It was certainly better than anything I was coming up with.

 

[FreeThinking]

@PeterDonis: Thanks. That makes me feel better about my confusion.

It's well after midnight here & I'm losing coherence, so I'll be brief. I think for someone like me who is studying on their own as a hobby with only the internet to ask questions of, the text becomes more important than it might be for someone who already understands it well. As to the equations, some are there but the intermediate steps are not. These textbooks are written for students much smarter than me, I can deal with that, but it would definitely be helpful if the authors would at least get the terminology correct and/or give more steps in the derivation. Their cavalier use of the terms "covariant" and "directional" derivatives have caused me no end of grief.

If you want to hear the whole, sad story, let me know. Otherwise I'll just leave it that, for me, the words really do matter when the intermediate steps of the derivation are completely left out.

Thanks again, Peter. I'll take your advice and not worry about it any further.

 

[FreeThinking]

@PeterDonis: P.S. Sorry, I forgot to mention that Carroll's lecture notes are practically burned on my screen, I've been studying them so long. And a lot of other printed & online resources. They've helped, but it has still be very difficult for a dummy like me.

Good night, and thanks again.

 

[Orodruin]

I have known many people who mix up the product rule and the chain rule nomenclature. MTW would not be the first nor the last. I have probably done so myself at some point in time.

 

[jbriggs444]

@fresh_42: Ok, now I understand you. I've never known the word "spleen" to mean that, but apparently it is used, at least in the U.S, I don't know about anywhere else, to mean "complain" or "rant". So, thanks for your guess. It was certainly better than anything I was coming up with.

I'd only ever heard the word used in that sense as part of the idiom "vent [one's] spleen". Apparently originating from a historical idea that the organ called the spleen is the repository of one's anger.

 

[Michael Price]

I have known many people who mix up the product rule and the chain rule nomenclature. MTW would not be the first nor the last. I have probably done so myself at some point in time.

I am surprised to hear that - they are quite different things and almost self-descriptive. I wouldn't even expect an undergrad to mix them up, let alone MTW.

 

[Orodruin]

I am surprised to hear that - they are quite different things and almost self-descriptive. I wouldn't even expect an undergrad to mix them up, let alone MTW.

I see it among undergraduates at a quite frequent basis (not in the first or second year, they still have it fresh). However, as has been mentioned here, the main thing is getting the maths right, not the English nomenclature. The product rule holds whether youcalk it the wrong thing or not.

 

[kent davidge]

youcalk it the wrong thing or not

was this on porpuse?

 

[PeterDonis]

it would definitely be helpful if the authors would at least get the terminology correct and/or give more steps in the derivation. Their cavalier use of the terms "covariant" and "directional" derivatives have caused me no end of grief.

What in particular is confusing you about those terms? Can you give an example of a usage of them that you find confusing from MTW (or another textbook if that's easier)?

 

[PeterDonis]

As to the equations, some are there but the intermediate steps are not.

Yes, that's true, textbooks will often leave out intermediate steps in the derivation, or assign them as homework problems instead of giving them in the main text. MTW does this fairly often. The homework help forums here can be useful if you're stuck on a particular problem.

 

[FreeThinking]

I see it among undergraduates at a quite frequent basis (not in the first or second year, they still have it fresh). However, as has been mentioned here, the main thing is getting the maths right, not the English nomenclature. The product rule holds whether youcalk it the wrong thing or not.

I agree that the main thing is to understand the maths, not the verbiage. This discussion is helping me on both counts.

MTW and every other book & article on the subject are what they are and we're stuck with them. My problem was, being such a novice especially with the non-coordinate symbols (I've been through a lot of this before using the old component-based contravariant/covariant tensors with focus on their transformation equations before I read Schutz's First course & discovered the "new" Cartanian way) I figured who am I to say that all these experts in the field are saying it wrong. But now that so many of you seem to think that's not such an unreasonable or arrogantly ignorant view to take, it tells me that perhaps I'm not as confused as I thought I was. I always try to avoid blaming my confusion on the expert authors since 99.99%+ of the time, I'm the one getting it wrong, not them.

Thanks.

 

[FreeThinking]

What in particular is confusing you about those terms? Can you give an example of a usage of them that you find confusing from MTW (or another textbook if that's easier)?

Thank you for that invitation, I'll take you up on it, but it will take me awhile to compose that post.

Now that I have finally figured out at least one way to derive Carroll's equation 3.71 on page 77 of his lecture notes, I'm going back and reviewing both Carroll & MTW to see if I understand things better now. As I do, I'll collect examples of where and how I got confused.

 

[SiennaTheGr8]

Product rule follows from the chain rule anyway.

 

[dsaun777]

MTW is probably not the best first textbook on GR to learn from. It is certainly comprehensive, but for that very reason it contains a lot of material that really isn't necessary if you are just trying to learn the basics of GR.

If you haven't looked at Sean Carroll's lecture notes on GR, you might give them a try:

https://arxiv.org/abs/gr-qc/9712019
They cover the basics, including the basics of the math--manifolds, tensors, differential geometry--in a much more focused way that might be easier as an introduction.

Those notes are brilliant for anyone wanting to start studying GR, Thank you. Anything similar for quantum field theory

 

[Michael Price]

Product rule follows from the chain rule anyway.

I don't find that proof terribly convincing. And it still doesn't excuse MTW mixing them up, which I am still amazed by.

 

[PeroK]

Product rule follows from the chain rule anyway.

Perhaps if you ignore mathematical niceties like:

The product rule requires the functions share a domain.

The chain rule requires the range of one function to be a subset of the domain of the other.

Function composition is fundamentally different from the simple product of functions.

If MTW had confused "proper" and "coordinate" time, I don't think he would have got away so lightly.

 

[SiennaTheGr8]

Perhaps if you ignore mathematical niceties

I do!

 

[PAllen]

Perhaps if you ignore mathematical niceties like:

The product rule requires the functions share a domain.

The chain rule requires the range of one function to be a subset of the domain of the other.

Function composition is fundamentally different from the simple product of functions.

It's perfectly easy to make the derivation of product rule from the multivariate chain rule completely rigorous.

 

[vanhees71]

It's fascinating, how one can get hooked up in a debate about a somewhat sloppy naming of a standard mathematical theorem known already in high school, as if there's nothing else of real interest in this great textbook on GR...

 

[fresh_42]

It's fascinating, how one can get hooked up in a debate about a somewhat sloppy naming of a standard mathematical theorem known already in high school, as if there's nothing else of real interest in this great textbook on GR...

The reason is: Some like the thought "Look, even the big ones make mistakes!" - not that they ever had claimed otherwise - and others jump into save their reputation - not that it would be necessary. And I have been fascinated by the mathematical questions whether the chain rule can be associated with chain complexes, or simpler, why the bilinear, associative $f\circ g$ as a multiplication instruction doesn't automatically show that the chain rule is an instance of the product rule. But I admit, wrong forum.

 

[vanhees71]

Yes, well, I wrote enough manuscripts to be very mild against trivial typos or just a bad formulation. The overall concept of MTW and the presentation of the material, however, is outstanding.