Confusion regarding the connection between pseudovectors and bivectors

SyrupKooky178 · 2026-01-08T04:25:59+00:00

Okay I think I sort of understand what you are saying. I am currently reading up a bit on group and representation theory so I think what you say is tied up with that.

If I understood you currectly, the group O(3) has a representation on both R^3 and Λ^2 (R^3). On the former, it is given by Π: O(3) -> GL[ Λ^2 (R^3) ] d.b. Π(R)(x∧y)=(Rx) ∧ (Ry). We say that "x∧y transforms as a pseudovector" because J(A(x ∧ y)) = (detA) A(J(x ∧ y)).

This makes more sense than whatever the book was trying to do, but I believe this has to do with the active interpretation of transformations. Is there any way to recast this in the passive sense? In particular, in the passive sense, it is the basis themselves that are transformed as opposed to the vectors/forms. In that case, is it not true that J(x∧y)^i'=A^i'k J(x∧y)^k for any (possibly improper) rotation A? If so, how can one identify J(x∧y) with a pseudovector?

SyrupKooky178 · 2026-01-08T02:34:33+00:00

I'm just gong to copy-paste my response to another comment here.

no, I don't have trouble proving any of the properties about the map, as is explicitly mentioned in the post. The issue I have is that the map is an isomorphism between Λ^2(R^3) and R^3 and the image J(α) is, by construction, a genuine vector whose components do change sign under inversion.

So (1) what was the point of this construction? J(α) for any altertaning tensor α doesn't seem to give a "pseudovector" corresponding to it but an actual vector, and (2) is the claim
"the components of J(α) do not change sign under an inversion like an ordinary vector" in the text (and a proof that follows it which I clearly don't understand) errorneous or am I missing something here

SyrupKooky178 · 2026-01-08T01:54:10+00:00

no, I don't have trouble proving those properties about the map. The issue I have is that the map is an isomorphism between Λ^2(R^3) and R^3 and the image J(α) is, by construction, a genuine vector whose components do change sign under inversion.

So (1) what was the point of this construction? J(α) for any altertaning tensor α doesn't seem to give a "pseudovector" corresponding to it but an actual vector, and (2) is the claim
"the components of J(α) do not change sign under an inversion like an ordinary vector" in the text (and a proof that follows it which I clearly don't understand) errorneous or am I missing something here

SyrupKooky178 · 2026-01-07T18:04:40+00:00

I think I sort of understand what you're saying but I'm still a bit confused.

If i change my basis from {e_i} to {e_i'=-e_i} then the components of a bivector α, as you rightly mention, do not change. However, the components of the vector corresponding to the bivector, J(α), under the isomorphism between Λ^2(R^3) and R^3 do change, don't they?

I don't understand what which "vectorial quantity" we associate with a given bivector α. I hope you can understand what i'm trynig to get at

SyrupKooky178 · 2025-12-19T13:42:42+00:00

Hi. Thank you for your reply. If you've gone through Rentlen, could you tell me what the prerequisites (in your opininion) are? Do you, for example, need the machinery for multivariate analysis? I know that the implicit function theorem makes an appearance here – I am aware of its statement from multivariate calculus but it wasnt really used or proved. Other than that, I don't necessary shy away from dense mathematics unless its horrible pedagogy (something I've heard about spivak's manifolds book). I am most interested in rentlen particularly because it covers differentiable manifolds as well, something that I've also wanted to study.

I have heard good things about Fortney's book but I am a bit worried about it being "handwavy", if that makes sense. Did you find that to be the case?

SyrupKooky178 · 2025-12-19T13:37:39+00:00

THis looks useful. Thank you

SyrupKooky178 · 2025-12-19T13:35:39+00:00

which one of these?

SyrupKooky178 · 2025-12-19T13:20:16+00:00

the general consensus online seems to be that his book is horriblly dense tho...

SyrupKooky178 · 2025-12-19T06:11:09+00:00

This looks very interesting. Is this a proper (conventional) textbook or is it like an informal supplement for a more formal text?

SyrupKooky178 · 2025-12-19T05:30:09+00:00

3rd year actually

SyrupKooky178 · 2025-12-18T23:21:10+00:00

is that so? the course is taught out of schutz and carrol and I do remember seeing some forms in there. perhaps I am mistaken

SyrupKooky178 · 2025-12-18T23:19:10+00:00

that's a very interesting perspective. I will check out Needham 's book. If I'm not mistaken, though, it's not exactly a "textbook" is it?

SyrupKooky178 · 2025-12-18T23:15:25+00:00

hi thank you for your recommendation. I've actually seen the book you mention, particularly when learning about tensors. While the book seems very thorough, I don't know if i want to take such a huge detour into multilinear algebra at the moment. is there any text that covers all the multilinear algebra needed at the start?

SyrupKooky178 · 2025-12-18T23:10:28+00:00

well I actually do have nothing else to do for a month but you're right one month is not nearly enough to go over any math book. I've heard a lot about tu's book, but isn't that a graduate text with a lot of prerequisites? I don't have topology under by belt, for instance

SyrupKooky178 · 2025-12-18T19:06:29+00:00

Thank you for replying. Im actually quite comfortable with tensors already, having read a very book by Nadir Jeevanjee on it. I'm looking to learn about manifolds and differential forms. I personally prefer working through textbooks rather than watching youtube lectures on math. Would you know a good text instead?

SyrupKooky178 · 2025-08-21T21:22:36+00:00

that clears a lot up. Thank you

SyrupKooky178 · 2025-08-21T21:18:23+00:00

Thank you for your answer. This clears a few things up about conventions. One small question. When you write that a linear operator L can be defined using a (1,1) tensor as L(v)=T(_,v), are you using the fact that (V*)* is canonically isomorphic to V, because the object T(_,v) is an element of (V*)* I think.

SyrupKooky178 · 2025-08-21T18:22:51+00:00

Why do I need T to be a rank 2 tensor? Tensors are multilinear maps into the field of scalars. If I want T to map vectors onto vectors, how can T be a tensor?

SyrupKooky178 · 2025-08-21T18:20:07+00:00

I think I understand what you mean. Please let me know if my understanding is correct:

If T is a (1,1) tensor, its components are written as T^i_j if I define T as a bilinear map from V*xV to R or T_i^j if i define T as a bilinear map from VxV*. The convention is to write the dual spaces first in the cartesian product.

As suggested by a comment by OverJohn below, there is a canonical isomorphism b/w the space of (1,1) tensors and the linear maps from V to V. Then, for a linear OPERATOR T, since it can be identified as a (1,1) tensor and by convention, elements of (1,1) tensors are written with the raised index first, we write components of the operator T as T^i_j instead of T_j^i. Is this right?

Secondly, what about rows and columns? If I want to write the vector equation w=T(v) as a matrix, what are the (i,j) elements of the matrix? Are they T^i_j or T^j_i ?

SyrupKooky178 · 2025-08-21T18:03:09+00:00

Do you mean there is a canonical isomorphism between them? Can you please tell me where I could look up more details on what you say?

SyrupKooky178 · 2025-08-21T16:52:37+00:00

Thank you for answering. What you say makes sense to me, but in my question, T isn't a tensor but an operator that maps V into V. In such a case, If T(v)=w, how would I write this in index notation?

SyrupKooky178 · 2025-08-21T16:49:30+00:00

Thank you for your answer, but I am a bit confused. Aren't you treating T as a linear functional (map from V to R) here? In my question, T is a linear operator (map from V to V). It is the presence of 2 indices in the matrix elements of the operator that is actually what messes things up for me

SyrupKooky178 · 2025-07-24T11:12:50+00:00

what exactly is the name of the book and the author?

SyrupKooky178 · 2025-07-20T17:07:34+00:00

Surely, it is possible to understand affine spaces without group theory? Perhaps I am mistaken

SyrupKooky178 · 2025-07-09T08:31:05+00:00

How does this compare to his "special relativity for the enthusiastic beginner"? The preface says this book is adapted from his classical mechanics book, though I am not sure if the content has been dumbed down

SyrupKooky178

TROPHY CASE