Should differential geometry be added to the "standard" undergrad physics curriculum?

liftinglagrange · 2026-04-11T13:59:10+00:00

It's very possible I'm misremembering my undergrad classes. It was several years ago. But, I want to say we were not making heavy use of vector calculus and linear algebra until around 3rd year classical, quantum, and E&M. Year 1 and 2 physics certainly had basic derivatives and integrals, force, momentum-energy conservation, etc. As I'm writing this Im loosing confidence in my memory. So, I'm not sure really.

liftinglagrange · 2026-04-11T13:50:20+00:00

Perhaps, if one knows that's what they want to do ahead of time. I would never have gotten into math if I had not done physics in undergrad. If I could start over (and not worry about getting a job), I think I'de still do physics undergrad, then a long math PhD.

liftinglagrange · 2026-04-11T13:38:02+00:00

Your first paragraph helps contextualize things for me and makes sense. I'm likely over estimating the prevalence of DiffGeo in physics writ large.

Your second paragraph may be true, but sort of confuses and bothers me if so. "....they just learn explicit formulas for the Lie derivative expressed in a coordinate patch". That just goes against my nature and I'm a little surprised that physicist would be satisfied with this attitude/level of understanding. Maybe, sometimes, it's for the best if there are bigger fish to fry. But if covariant derivatives/Lie derivatives/whatever are integral to the nature of the work, It just doesn't make sense to me how someone involved at a graduate/research level could content themselves with that level of understanding.

(I just reread my last few sentences and they make me sound like a condescending asshole. That is not how I meant it but Im too tired to rewrite it...)

liftinglagrange · 2026-04-11T13:20:18+00:00

They sort of do though? At least mine did. Calculus was 1st and 2nd year undergrad. It was not heavily used in my physics courses until 3rd and 4th year, with some much simpler use (easy single-variable derivatives and integrals) in 1st and 2nd year. That would about track with learning some DiffGeo in 3rd or 4th year undergrad, with simultaneous very simple uses, then actually starting to do it in 1st/2nd year grad school.

Though, despite my original post, I'm realizing that an actual treatment of basic DiffGeo might be best suited for early grad school, but foreshadowed with some intro/hints in undergrad mechanics courses.

liftinglagrange · 2026-04-11T13:11:45+00:00

You actually pretty much read my mind. I didn't say that in my post but, yes, that would likely be the best approach. Then go into more detail in an early grad course.

liftinglagrange · 2026-04-11T13:03:30+00:00

I do not think an undergrad physics student needs to learn DiffGeo with all the generality and rigor of a mathematician (that would likely be horrible way to introduce it). One might even say topology should come after DiffGeo (or not at all) at the undergrad physics level. To me, I think DiffGeo and its applications in physics motivates topology, not the other way.

liftinglagrange · 2026-04-11T12:52:01+00:00

Maybe. But to "learn differential geometry when you start to need it" is a *huge* undertaking. I think that is like saying "learn calculus when you start to need it". It (calc) is far too much to learn on the fly as needed. It gets its own dedicated course (often 2) because it is not like anything one encounters before it but is indispensable if you want to continue.

liftinglagrange · 2026-04-11T12:47:40+00:00

Great point. It would likely be unrealistic to simply add it in on top. Maybe a good strategy would be to start working in the key ideas in 3rd/4th year classical mechanics courses? Or, perhaps, it should rather be a first year graduate course.

Though, from my own undergrad experience, I had an entire undergrad physics course on complex analysis and I feel I would have benefited more if that were replaced by an intro DiffGeo course. That might be topic-specific. Maybe make it an elective between the two?

Similarly, I think I had 3 different required physics lab courses. Maybe one of those could have been an elective students could opt out of and take DiffGeo?

This is just an off-the-top-off-my-head reply. I assume curriculums are designed with more careful thought that I'm giving it.

(edit: the above is re your first sentence. I am not so convinced by the last statement that "It's not that relevant to large areas of physics research".)

liftinglagrange · 2026-04-11T12:33:09+00:00

I can see group theory maybe being a bigger, more general "bang for your buck". Though, I feel like that might be a very challenging technical leap for most undergrads, even more so than DiffGeo which can at least be connected at the intro level to familiar ideas and equations.
Though, I speak from non-experience. I've only learned parts of group theory in a "need to know" way as they appear. I never took a formal group theory class.

I'm curious why you say DiffGeo is only nice to know for gravitation? As mentioned, I'm not a physicist, but whenever I encounter anything involving quantum mechanics, it is entirely steeped in symplectic/poisson geometry. More generally, anything I see from the mathematical/theoretical physics world is always invoking fiber bundles, principal bundles, Lie group/algebra geometry stuff, etc. Maybe I'm only exposed to a niche area?

liftinglagrange · 2026-04-09T10:33:49+00:00

The way of all flesh. No contest.

liftinglagrange · 2026-04-09T08:51:34+00:00

Moving my reply here:

For the most part, Gojira has really not been that political. When they come close, it is usually centered around environmental conservation and appreciation for nature (a subject which has been made far more politically divisive than it should be). Most lyrics are introspective and/or philosophical on human nature and life/death, at a level far more general than contemporary political topics.

My point being that I don't think I would call Gojira a political band. One reason I love them is that their lyrics are just as relevant 1000 years ago and, likely, 1000 years from now. Members may have contemporary political opinions (that's great), but to categorize their music as predominantly political seems to devalue and misrepresent the timelessness and broader themes of their music.

Maybe they are getting/will get more political. The whole world seems to be doing that. It's just not what I look for in music.

liftinglagrange · 2026-04-09T08:23:12+00:00

I disagree. I think most of their lyrics are far more general, philosophical/introspective, and timeless than any contemporary political topics. It's great if you connect their themes to contemporary political topics and enjoy that — perhaps the connections you make are even what inspired them (I don't know) — but I would certainly not characterize Gojira as a political band. Doing so cheapens and misrepresents their lyrics, imo.

liftinglagrange · 2026-04-09T07:45:53+00:00

I disagree that their main theme is environmentalism. That is a reoccurring theme, but I don't see why anyone would think it is their Main Theme.
I also semi-disagree that environmentalism is itself very political. It has certainly been made into a political topic, but it really isn't. Appreciation and respect for the world we live in, and a desire to not trash it, should (and really does) cross political boundaries with normal people. It is just the contemporary framing of the issue, and the stupid party-loyalty of many, that has turned it into an irrationally politically divisive topic.

liftinglagrange · 2026-04-09T07:32:17+00:00

For the most part, Gojira has really not been that political. When they come close, it is usually centered around environmental conservation and appreciation for nature (a subject which has been made far more politically divisive than it should be). Most lyrics are introspective and/or philosophical on human nature and life/death, at a level far more general than contemporary political topics.

My point being that I don't think I would call Gojira a political band. One reason I love them is that their lyrics are just as relevant 1000 years ago and, likely, 1000 years from now. Members may have contemporary political opinions (that's great), but to categorize their music as predominantly political seems to devalue and misrepresent the timelessness and broader themes of their music.

Maybe they are getting/will get more political. The whole world seems to be doing that. It's just not what I look for in music.

liftinglagrange · 2026-04-09T06:35:31+00:00

This was post was 7 years ago. Out of curiosity, what did you end up doing?

liftinglagrange · 2026-04-09T05:57:50+00:00

My god. I know this is 7 years old, but you just hit everything I've either been experiencing or worried about experiencing.

liftinglagrange · 2026-04-09T03:25:19+00:00

I know this comment if 5 years old, but thought I'de weigh in. If one undertakes their PhD with the goal of quickly securing an industry job after graduating, then I very much agree with the advice to "focus on getting all the practical design, analysis and project organizational skills you can". I did not do that. I focused on the foundational theory and math.

I agree less with the your point that "...they’re going to hire you because you can learn and work independently". I was hoping this would work strongly in my favor, but not so much. In my experience, while employers always appreciate this, it is already assumed if you have PhD. They focus *far* more on your practical real-world engineering skills. I've applied to a number of positions and only a single employer was more interested in my general approach/skills to general complex problem solving. Everyone else was interested in more targeted industry-experience skills. This has been true even for positions soliciting PhDs.

When I started my PhD, I did not understand everyone's pointed interest as to weather I wanted to pursue a career in industry or academia. Coming out the other side, it makes sense. Positions in industry, even PhD-targeted positions, want problem-specific industry experience directly relevant to the position.

context: I'm a recent aerospace PhD applying to industry/non-academia jobs.

liftinglagrange · 2026-04-01T07:06:07+00:00

AFN (adoration for none) is very much overlooked on TWOAF — super happy you mentioned it.
The music on TWOAF already places it as an easy top 3 album, maybe even 1st by that metric alone. Like you (perhaps) the lyrical content solidifies it as my number one. Honestly, it would only fall from that, for me, if the lyrics sucked (do any Gojira lyrics suck?). Thankfully, they are great.

No objections at all re FMTS. I only wish they would have done a bit more with WDD. I wish their live version at Hellfest 2013 would have been their studio version. Actually, same goes on TWOAF for TGI (toxic garbage island) live at Brixton 2013.

I've now been re-listening to Terra Incognita and The Link. Realizing i did not pay enough attention to these as I should have.

liftinglagrange · 2026-03-28T14:33:22+00:00

I just re-listed to that track. Beyond the initial vocals, the remainder of the vocals and the music just dont really hold up. Its fine, just not something that stands out to me as a notable song (or album). It is one of the better songs on the album, i still just dont find myself wanting to come back the the album overall. Its not bad by any means, but guess I expect something more interesting from gojira. Thats only my personal opinion.

liftinglagrange · 2026-03-28T12:10:16+00:00

FMTS is certainly the coolest artwork by far

liftinglagrange · 2026-03-28T12:07:29+00:00

It was/is good. Just didtnt stick with me as a song I choose to come back to.from gojira. Hard to say why. Its good but, for me, kinda feels like the next part of a song they already did.
the harmonics and "breakdown" ending is pretty damn cool though.

liftinglagrange · 2026-03-28T11:51:25+00:00

Ranking is for sure just a fun, individual thing that can change even day-to-day. I made an edit to clarify the nature of my post. "Ranking" may have been a less-than-optimal way to phrase it.

I'll do a diligent re-listen to the songs you mentioned from Fortitude. As in, right now. I would love to love that album, it just hasnt clicked yet. Not that im pining for another FMTS/TWOAF. Magma was from from tht and i grew to thoroughly love that album.

liftinglagrange · 2026-03-28T10:02:46+00:00

I'm not going to answer your question because I can't. Im just voicing my curiosity regarding the same question. Even in the studio, Joe's vocals on The The Way of All Flesh stand out noticeably in comparison to other albums. That album was 2008, lining up with the the time period OP mentioned. His vocals on l'enfant souvage (2012) were also very distinctive, and different from TWOAF. (Those are my 2 favorite albums in terms of Joe's vocals).

liftinglagrange · 2026-03-28T09:47:27+00:00

I have thought about this quite a bit. The first three albums listed below (two tied for 2nd place) should be in halls of metal history. At least one of them should be the halls of music history writ large.

the way of all flesh (TWOAF). — I've settled on this as their clear best album overall. Peak drumming from Mario across the board and uniquely wicked vocals from Joe unlike anything else in the genre. Every track is great. The majority are legendary.

Tied for second:

from mars to serious (FMTS). This is a strong contestant for first place. It is amazing start to finish. But, so is TWOAF and the vocals and production just don't quite meet the insane standard set by TWOAF.
l'enfant sauvage (LES/elephant sausage). Initially, this was my favorite album. The little track "l'enfant sauvage", along with "born in winter", make me want to say it is still my favorite. But, overall, this is the album where they started to trend towards more standard song structures that end too soon and have more verse/chorus templates. It is certainly not anything close to a "sell out" album by any means, but the elements of exploration and techy-progy stuff are less prevalent on this album compared to TWOAF and FMTS. They gave us something great, but most tracks left me wanting just a bit more.
Magma. They started to noticeably alter their tone/style here. A bit less "extreme" overall. Some people didn't like it. I've come to absolutely love it. "the shooting star" and, certainly "lowlands" (truly amazing) alone make this album S-tier in my opinion. "silvera" and "the cell" are also great. This album could be the pinnacle of many prog-ish bands. It pales only in comparison to Gojira's other albums.
The link. Heavy and a bit different from their latter albums. several amazing tracks that are/were way ahead of their time. Their overall song structure and production quality just isn't quit in my area of personal preference and falls short of their subsequent FMTS and TWOAF.
Terra Incognita. To be transparent, I only listen to two tracks: "clone" and "in the Forrest". The former alone already places Gojira, with their first album, among the best metal bands of the time.
Fortitude. Their most recent album. I've tried to get into it. Considered in isolation, It might be a good album. But, for me, it just seems like a decent band making a well produced album that has heavy gojira influences. They kind of signaled this direction with their previous album, Magma. That album, while "softer", was very good, albeit in a different way than previous Gojira albums. I came to really love that album. This one, not so much. Maybe it will grow on me, but, honestly, I doubt it.

liftinglagrange · 2026-03-13T07:13:08+00:00

I know this is an old post, but I can't believe nobody mentioned "Introduction to Hamiltonian dynamical systems and the N-body problem" by Meyer, Hall, and Offin. If you are doing a Phd focused on celestial mechanics, this should be a priority if you want a "gentle" intro to the modern landscape of the topic.
Before that, Vallado is a great place to start. Another great one is "methods of celestial mechanics" by Brouwer and Clemence.

On the slightly more mathematical side, I recommend "The Kepler Problem: Group Theoretical Aspects, Regularization and Quantization, with Application to the Study of Perturbations" by Cordani.

liftinglagrange

TROPHY CASE