the basis vector conundrum REVIEWED

The Basis Vector Conundrum PhL 4.5.16 5.16.16: This is where I realized that I had to change my entire wedge doc from ei to ui basis! That change was a major effort but things are very much better for it! Wedge doc has ground to a halt because there is a major foundational problem, and this problem may exist as well in tensor doc. I am now putting on the hat of a critic reader who is trying hard to absorb all the information I am presenting, but who is confused. Here is what that critic reader says in a series of "points": The Critic Reader Speaks 1. I understand that, in your x-space/x'-space transformation picture with Cartesian x-space, you think of the un as your "axis-aligned" basis vectors, u1 being with the x axis, u2 with the y axis, and so on. And I understand that in this picture, you have defined en to be "tangent base vectors" which form a different set of basis vectors for x-space. I also understand that your transformation rule for vectors is v'i = Rijvj where you have two kinds of components, those of the vector v in x-space, and those of the vector v' in x'-space into which that vector is transformed. Yes, so far this all makes sense to me. I also understand that in this notation, you can "expand" a vector either as v = viui or as v = v'iei and that these two expansions must necessarily have different "coefficients" that they are exactly as just stated. I understand that it is your intention to treat "tensors" as objects that can be "expanded on basis vectors" just as you have done above for vectors. I understand your desire to have such expansions be valid for "arbitrary basis vectors" and not just the simple ui basis vectors of Cartesian space. I understand that if you want to maintain the "tensor structure" of things, you must write M = M'ij eiej as the "correct" expansion of a rank-2 tensor M onto those tangent base vectors, and that you must therefore have a prime on M'ij so that it is the x'-space set of tensor components. I like your idea of associating "arbitrary basis vectors" en with the tangent base vectors of some appropriate underlying transformation which is in effect custom-created for those arbitrary en. This then allows you to do expansions like M = M'ij eiej and to be able to claim that M'ij are the components of a true tensor, rather that to be just some generic "expansion coefficients". I also understand that in your "tensor algebra" discussion, you have vi = ui v as your definition of "the component of an x-space vector" and that such a component is a "component of v in the u basis", and that such a component could be written v(u)i in a more elaborate notation than you use in your Chapter 2. I understand that this notion of a "component" affects almost everything in early Chapter 2, including the meaning of the components on things like g'ij and (en)i . In other words, I understand that your entire "tensor algebra" is based on "objects" which have components defined relative to those "axis-aligned" basis vectors un. 2. Regarding your Chapter 10: I understand that you have a Chapter 10 on the subject of "differential forms" , and that in this chapter you set up a transformation between two spaces and that you want this "transformation picture" to align with your Chapter 2 and Tensor Doc notions of a transformation between two spaces, so that you can leverage off all the work you have done in the general area of transformations. In particular, in Chapter 10 you want to have the axis-aligned basis vectors like u1 in x-space map into "tangent space vectors" like u'1 in x'-space, and that this then provides a perfect "scenario" for understanding the business of manifolds in x'-space. Thus, I see that this is a real-world application of the notion of your "tangent base vectors", and that this gives a good justification of including the Chapter 2 tensor algebra review in this wedge document of yours. I understand in this Chapter 10 how useful it is to have a "good handle" on the four kinds of basis vectors en, e'n , un and u'n all of which you have clearly defined in Chapter 2. In passing, I note in your Chapter 10 presentation that you need to make use of the following two dual space basis elements: λi ≡ <ui| = basis functional in dual x-space i = 1..n // ui axis-aligned λ'i ≡ <e'i| = basis functional in dual x'-space i = 1..m // e'i axis-aligned . (10.7.12) and that both of these objects involve axis aligned basis vectors and not "general arbitrary" basis vectors. So it occurs to me that in your discussion of this λi object, you might not have to treat the case of "general" basis vectors which you think of as <ei| . 3. Now we come to your general exposition of tensor products and wedge products in Chapters 4 and then the carefully arranged set of Chapters 5,6,7,8. I understand that in the differential forms Chapter 10 you are going to be dealing with "wedge products" and that is one reason to have these detailed chapters 5,6,7,8 presented in such a highly systematic and parallel fashion so everything is clearly described and laid out for readers like me. I understand your desire in Chapters 4-8 to "work with general basis vectors" we can call en rather than just with the somewhat limited set un. I understand that you want to present things in the most general manner possible, within reason. I understand that one of your options in Chapters 4-8 is to think of those "general en" as being tangent base vectors for an artificially conjured-up transformation derived from those en , and that this then allows you to claim that your tensor expansion coefficients like this in M = M'ij eiej are in fact true tensor components. I see that this then requires you to put a PRIME on many objects which appear in these chapters such as M'ij and such as v = v'iei and I see that "vectors" appear a lot in these chapters. So I understand that this causes a "proliferation of primes" occurring in these chapters. I understand that another option for these chapters is to give up on having the en be "general basis vectors" and you could restrict them to be the axis-aligned un . I understand that in this case, expansion coefficients then do not need primes, but they still have the desired tensor interpretation which I know is important to you to maintain. In this option, you don't have to construct a custom transformation, because the tensor components are all x-space objects and could then apply to any transformation. I see that as a pro of doing this option, whereas a con is that the basis vectors are then not "general". I do point out that in fact you may have already in fact selected this second option! For example, in Chapter 5 you say (just a sample equation) (ei ei ..... ei)jj...j = (ei)j (ei)j .... (ei)j = δij δij .... δij . (5.1.4) Right here by saying (ei)j = δij, you are saying that your en are in fact "axis aligned" basis vectors! So maybe what you have done is really used the un and have just called them en. 4. So let's look more closely at this last point. I know you had in mind that, in saying (ei)j = δij, you were meaning (ei(e))j = δij where the component index j is associated with the e basis, and that you were just economizing on notation by writing this as (ei)j = δij. So you were using (ei)j = <ej|ei> and more generally vj = <ej|v> to define your "components" in Chapters 4-8. In light of this situation, I the reader than ask you the author two questions: Q1: Regardless of whether your notation is (ei)j = δij or (ei(e))j = δij, it seems to me that you are then assuming that your basis vectors en are not general vectors, but in fact are specialized to be axis aligned. It is this δij expression that is your own definition of "axis aligned". I guess this is a comment rather than a question. I thus challenge your claim that Chapters 4-8 are written for general en vectors. Q2: If you DO intend to define "components" by vj = <ej|v> where en are "general" basis vectors, then almost all of your Chapter 2 is not longer applicable because there you had vj = <uj|v> for every index on every equation. This it seems to me makes your document very inconsistent. The Author Responds The comment and question are both very helpful. The Q2 item is rather frightening, to have two separate worlds which don't connect. I will address the Q1 item first. Q1 Response: First of all, when I write (ui)j = δij I do NOT mean that x-space is Cartesian. What I really mean is that (ui(u))j = δij . What that in turn means is this: "components of vectors are taken relative to the u basis, whatever that basis might be". I noted elsewhere that (ui(u))j could also be written (uj)(u)j where the (u) is a marker on the component index j, saying that this is a u-basis type component. To say that (ui(u))j = δij is equivalent to saying that you can expand ui = Σj(ui(u))juj because then you get ui = Σjδijuj = ui. But then to be consistent you need to expand ANY vector in this same manner, so then v = Σjvi uj where the previous situation was the special case v = ui. So I think I have had a wrong intuition about the notion of "axis-aligned basis functions". You could take any basis b and say (bi)j = δij and v = Σjvi bj . So here is a correction: Fact: To say that (bi)j is an "axis-aligned basis vector with (bi)j = δij" is merely to say that you are going to use the b-basis for the expansion of vectors, v = Σjvi bj, so that the meaning of a component index j on vj is that vj = <bj|v> . If you like, (bi)j = (bi)(b)j = δij . Example: In tensor doc and wedge doc I define the ui vectors to be (ui)j = δij as if this meant the u basis was some "special basis". It does not mean that. It just means the I am going to be defining component indices to be relative to this ui basis, and therefore all my tensor equations which involve components are "relative to this u basis". All tensors like Mij and gij are "relative to the u basis". There is nothing wrong then with defining any other basis en . But your tensor components are always with respect to the ui unless you change your whole underlying foundation. Apology: I was thinking that describing a basis en as (ei(e))j = δij was somehow different from the basis un as (ui(u))j = δij. If someone gives you two physically different basis vectors un and en , you can use either (ui(u))j = δij or (ui(u))j = δij . But these then imply two different meanings for an object like vj , and that seems very dangerous. In one meaning you have vj = <ej|v> and in the other meaning you have vj = <uj|v> and for a fixed physical vector v these will be different numbers. !!! This last statement then reflects on my critic's Q2 item. Fact: You simply cannot write a document with two different meanings for the number vj ! A document has to be consistent. Conclusion: I really should change all my e references to u references in Ch 4,5,6,7,8. This then also solves the problem of having primes on components, which is a big mess, and it solves the problem of having to create contrived transformations so you can say v = Σiv'i ei . Remember that you cannot have a tangent base vector without a corresponding transformation! I think THIS is the way out of the conundrum. Example: In x-space I chooise (ui)j = δij which just means vi's meaning is v = Σivi ui In x'-space I chooise (e'i)j = δij which just means the meaning of v'i is v = Σiv'i e'i . Both these expansions agree with my summary of all expansions: 1 V = Σn Vn un where Vn = un V = <un|V> 2 V = Σn Vn un where Vn = un V = <un|V> 3 V = Σn V'n en where V'n = en V = <en|V> 4 V = Σn V'n en where V'n = en V = <en|V> 5 V' = Σn V'n e'n where V'n = e'n V' = <e'n|V'> 6 V' = Σn V'n e'n where V'n = e'n V' = <e'n|V'> 7 V' = Σn Vn u'n where Vn = u'n V' = <u'n|V'> 8 V' = Σn Vn u'n where Vn = u'n V' = <u'n|V'> (2.7.1)