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I do!Perhaps if you ignore mathematical niceties
I do!Perhaps if you ignore mathematical niceties
It's perfectly easy to make the derivation of product rule from the multivariate chain rule completely rigorous.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.
The reason is: Some like the thought "Look, even the big ones make mistakes!" - not that they ever had claimed otherwise - and others jump in to 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.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 highschool, as if there's nothing else of real interest in this great textbook on GR...
That moment when you are going to explain to your students that a time integral is missing in a course book and show the ”correct” version in your own book just to realise that - although you remembered the dt unlike the other book - the integral sign with the limits is missing from your book as well ...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.
I have forgotten to mention the "internet effect". I remember a thread which I thought would have been instantly closed or at least after a few answers. It felt to have lasted internally. The question was, whether zero is a real number or some nonsense like this.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.
Ok. That's why I asked the question, rather than assuming they were wrong. That's the kind of thing I meant when I asked, "What am I missing?" So the two rules are connected in this way. I just never spotted that in all my Googling.Product rule follows from the chain rule anyway.
... 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.
I want to be clear that, despite how it may sometimes sound, I am in no way, shape, or form blaming MTW or any other author or book for my difficulties in understanding this subject. Every reader/student has a different level of background, understanding, and skill. Every author must decide for themselves who their target audience is. I just happen to be on the outer fringes of the target audience of MTW, so their book is very challenging for me. It is simply a fact, not a complaint, that MTW, and others as well, seem to have used the terms "covariant derivative" and "directional derivative", as well as other terms, in a way that has confused me. I am not talking about the occasional typographical mistakes, but only what appears to be deliberate uses.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)?
If I did not agree, I would never have spent so much time trying to understand it. I have found no other book that covers as much material, even if the coverage is a challenge for the likes of me. Well, it keeps me off the streets at night.The overall concept of MTW and the presentation of the material, however, is outstanding.
Ok, I've stared it a while longer, and here's what I'm seeing:Also, starting with the last two paragraphs at the bottom of page 208, we establish that ## \boldsymbol {e}_\beta ## and ## \boldsymbol {\omega}^\alpha ## are general bases dual to each other. Continuing onto page 209, equation 8.19a says that ## {\boldsymbol \nabla}_\gamma \equiv {\boldsymbol \nabla}_{{\boldsymbol e}_\gamma} ## . Then further down the page, equation 8.20 defines ## T^\beta_{\alpha,\gamma} \equiv {\boldsymbol \nabla}_\gamma T^\beta_\alpha \equiv \partial_{{\boldsymbol e}_\gamma} T^\beta_\alpha \equiv \partial_\gamma T^\beta_\alpha ##. With a general basis, not a local Lorentz frame, why are we defining the directional derivative ## {\boldsymbol \nabla}_{{\boldsymbol e}_\gamma} \equiv {\boldsymbol \nabla}_\gamma ## to be a partial derivative ## T^\beta_{\alpha,\gamma} \equiv \partial_\gamma T^\beta_\alpha ##? If we were using a coordinate basis, say ## \left \lbrace {\boldsymbol {\xi}_\gamma} \right \rbrace ##, it would make sense since ## {\boldsymbol {\xi}_\gamma} \equiv {\boldsymbol \nabla}_{{\boldsymbol e}_\gamma} ##, the directional derivative operator along the coordinate curve ## {\boldsymbol {\xi}_\gamma} ##. Perhaps if we stare at this section long enough, it might dawn on us what they actually mean. ... I tried to work through equation 8.19a & 8.20, but I'm still not getting the same result they seem to get.
I don't understand what you're doing here. The Lorentz transformation ##\Lambda## doesn't appear anywhere in the section of MTW you're referring to, and anyway you don't use a Lorentz transformation to transform from local inertial coordinates to general curvilinear coordinates.Based on how MTW defines things, as described above, I get##\boldsymbol \nabla_\gamma T^\beta_\alpha = \Lambda^\mu_\gamma T^\beta_{\alpha,\mu}## , using ##\boldsymbol e_\gamma = \Lambda^\sigma_\gamma \boldsymbol \xi_\sigma## where ##\boldsymbol \xi_\sigma## are the coordinate basis vectors.
I don't understand what you're doing here either. It doesn't help that you're throwing in your own notation ##\boldsymbol \xi_\gamma##, which doesn't appear anywhere in MTW. MTW always uses ##\boldsymbol e## for the basis vectors, not ##\boldsymbol \xi##.but if I replace the ##\boldsymbol e_\gamma## with ##\boldsymbol \xi_\gamma## , I get ##\boldsymbol \nabla_\gamma T^\beta_\alpha = T^\beta_{\alpha,\gamma}## which seems to be what MTW says it should be.
You are confused. You don't apply the ##\boldsymbol \nabla## operator to the components of a tensor.MTW has used ##\boldsymbol \nabla## is such a way that it generates gamma correction terms when applied to a general tensor. But applying it to just the components of a tensor does not generate those components unless we interpret it as the semicolon operator, which they do not seem to do in (8.20).
I don't know where you're getting this from. You don't use a Lorentz transformation to go to general curvilinear coordinates. See above.MTW just defined ##\boldsymbol e_\beta## to be a general basis, not necessarily a coordinate basis. Yet in (8.20) the ##\Lambda^\sigma_\gamma## needed to define the general basis is nowhere to be found
Have you encountered covariant derivatives in other textbooks? Have they confused you there?This is a case where the math itself confuses me even if we ignore the text.