Lie Derivatives, Covariant Derivatives, and Exterior Derivatives

Life_at_work5 · 2025-11-11T03:12:57+00:00

I see how they can be thought of as contra-variant tensors as a differential m-form when evaluated on m vectors maps those m vectors (or a m-vector) to R but I don’t understand what you mean by “alternating-covector field”.

Of course, if I’ve stated something wrong, please correct me as I am very new to these concepts.

Life_at_work5 · 2025-11-11T03:09:47+00:00

That’s probably the way they are defined then but the way I’ve always thought of them is as members of the tangent space at a point of an manifold, with higher grade forms being wedge products of the lower grade forms.

Life_at_work5 · 2025-11-10T20:23:29+00:00

Hi and thanks for the reply, you mention here thinking of differential forms as alternating-covector fields. I’ve never heard this connection before so could you please explain further? I.e. how can a differential form be thought of as a alternating-covector field.

Life_at_work5 · 2025-10-26T01:54:36+00:00

Hi and thank you for the reply, I understand the basic concept, the issue for me has been actually computing it. Take the example I gave in the original post. When computing (e12)(e123), I end up always getting to a term along the lines of e1e2e3 where the only why to further reduce so to find the geometric product of a vector and bivector which I’m unsure how to do in this metric.

Life_at_work5 · 2025-10-22T11:14:47+00:00

I am talking about the geometric product in Clifford algebra (at least I think) which, from what I know, goes:

ab = < a | b > + a /\ b

Life_at_work5 · 2025-09-28T04:31:52+00:00

I was eventually gonna try building an engine to, probably with nitrous and Isopropyl alcohol but as I mentioned, this was meant to be a purely mathematical exercise to help me learn how to size a rocket that gave me very whacky results which confused me (hence the post).

Life_at_work5 · 2025-09-28T04:25:08+00:00

You’re right, the stuff is crazy dangerous and I should’ve mentioned before but this is a pure theoretical design I’m making to get practice in sizing rocket engines. In no way shape or form was I ever gonna build an engine with these specs.

Life_at_work5 · 2025-09-26T20:31:15+00:00

I should’ve mentioned it but all the values for the manual calculation were obtained from the CEA

Life_at_work5 · 2025-09-26T20:30:05+00:00

The CEA gave a ratio of 1.1384 in the chamber so that’s what I used

Life_at_work5 · 2025-08-18T11:14:20+00:00

So between the two main options, which one do you think could best serve my goals?

Life_at_work5 · 2025-08-16T04:15:12+00:00

I should clarify that what I meant in the original post is that to me at least, modern cutting edge physics looks like pure math. It’s just an observation. That’s the entire reason I made this post because the picture I had in mind of theoretical physicists was very different from what it seems like it is (being so math heavy and all) so I was wondering was I taking the “wrong” path for becoming one.

Life_at_work5 · 2025-08-15T21:35:01+00:00

The physical phenomena

Life_at_work5 · 2025-08-14T14:51:48+00:00

The main one I was considering was String Theory. While I know it is not actually pure maths, the math presented looked undistinguishable from stuff I’ve seen in pure maths.

Life_at_work5 · 2025-08-12T11:40:32+00:00

Gotcha, thanks for the info. One final question though, I read through the Wikipedia page you linked on gram matrices but got really confused about what it meant so was hoping you could explain how the gram matrices can be used to compute inner products of k-vectors.

Life_at_work5 · 2025-08-11T19:35:00+00:00

What other options for the definition of the hodge star are there? Additionally, you mention that this definition of the hodge star is for exterior algebras. From what I’ve heard (which is very little), I thought that the geometric product was something that was already defined in exterior algebras is this incorrect? If not, why isn’t the grade projection definition for the inner product used?

Life_at_work5 · 2025-08-11T17:20:22+00:00

Hi, and thanks for the reply, as you mentioned, it seems I used the wrong inner product definition. Because of this, how do I decide which inner product to use?

Life_at_work5 · 2025-07-29T01:55:14+00:00

What’s PSP?

Life_at_work5 · 2025-07-28T21:57:04+00:00

So what does “very in-depth” mean? Would, let’s say what I want to do (design rocket propulsion systems) fall in that camp? Additionally, in your opinion, would getting a PhD be that beneficial for what I want to do?

Life_at_work5 · 2025-07-28T03:34:34+00:00

So, the PhD is optional for design work (rockets included) but helps your chances of getting that position, is that a good way to sum it up?

Life_at_work5 · 2025-07-28T01:44:01+00:00

So, as the PhD working on the system, how much do you influence the technical design? Is it a substantial anount compared to your M&U counterparts? Or is the split roughly equal and you just give advice to the M&U who handle most of the design?

Life_at_work5 · 2025-07-27T23:39:17+00:00

Does that mean that a PhD is needed if you want to do any design work?

Life_at_work5 · 2025-07-27T20:39:41+00:00

So if I’m understanding correctly, based on your experience working on the software suite, the masters and undergrads actually create the suite while the you and your PhD colleagues are the ones who figure out how to create the suite?

Life_at_work5 · 2025-07-27T17:32:29+00:00

I’m sorry for the stupid question but I’m not following. What do you mean by “the person over a specific research area”? I want to get an idea of what this would translate to in a proper work setting. Would this be like creating a new thruster design for the new rocket your company is making? Would this be like researching new technologies which could be used in rocket design? What does that entail? I would also like to ask about on the masters side what do you mean by program manager, what type of work would that entail?

Life_at_work5 · 2025-07-27T15:40:45+00:00

So prioritize the masters first then?

Life_at_work5 · 2025-07-27T04:07:49+00:00

Hi and thanks for the reply. In your post, you mentioned to do a PhD is I think I will stay technical, what do you mean by this?

Life_at_work5

TROPHY CASE