tensor questions 1

Debug Doc PhL 10.7.11 Ask questions, answer questions. [ In this doc I asked what at the time seemed like pretty difficult questions. I had lots of words and clutter and examples, but now I feel these questions are all resolved. So the clutter is removed, and I have just left the questions and what I think are the right answers. ] ______________________________________________________________________________ Question #1. What can be said about the "contravariant" nature of the vectors en and 'n ? "The pair" { 'n , en } are indeed connected by 'n = R en, and thus I would say that en transforms as a contravariant vector into 'n with respect to F. I would NOT say that en was a covariant vector in this same regard.. " [ In retrospect, I now think of en for sure as a contravariant vector. Is en contravariant-by-definition? Yes it is. We define it according to en ≡ S'n so that 'n = R en. I think I have resolved this question in my section on "contravariant by definition". ] _____________________________________________________________________________ Question #2: Is it meaningful to ask if a vector V is contravariant? [ Yes it is. You consider V' ≡ RV and if V' does not conflict with an existing vector, the answer is yes, your V is contravariant. That is to say, if it is not already known to be contravariant or known not to be, you try to force it to be and see what happens. ] _____________________________________________________________________________ Question #3: Suppose V is the x-space end of a contravariant pair {V ', V} under F, so that V' = RV. Suppose x-space is Cartesian. How would you expand vector V on "basis vectors" so that the components Vi and V'i were the components involved in V' = RV ? Possible Answer # 1: You would write V = V1 + V2 +... where the basis vectors are the axis aligned unit vectors in x-space. Possible Answer # 2: You would write V = V1 e1 + V2 e2 +... [ The correct answer is #1, and I have stated this in tensor.doc right where vector types are defined. This is the correct answer even if g ≠ 1. ] _____________________________________________________________________________ Question #4: Do we ever use things like en when x-space is non-Cartesian? [ Answer: we can certainly define them and have them exist in general Picture A. I am not sure how one might use them, but one could use them. ] _____________________________________________________________________________ A brief collection of possibly useful ideas and one remaining puzzle. [ merged in from another doc.] Now that my first full draft of Tensor Doc is finished, after very many days, questions still remain, general puzzlements shall we say. _____________________________________________________________________________ A rank n tensor is an object with n indices which transforms just as would the outer product of a set of n vectors having the same type of indices. So things really are tied to the notion of "vector: and there are two types of vectors. _____________________________________________________________________________ In special relativity, we have G = RRG which says that a transformation of a vector V' = RV must preserve the length of a vector as defined in terms of G. Because G is not Cartesian, there are the two kinds of vectors existent. _____________________________________________________________________________ Is a global x' = F(x) always present somehow? _____________________________________________________________________________ In the special relativity application V' is a 4-vector and 4-vectors are associated with the 1/2 1/2 representation of the Lorentz group, and so all tensors transform in a manner which is based on this vector representation. _____________________________________________________________________________ One item is the idea of up and down indices on objects like xi which are neither contravariant nor covariant vectors. Such indices can be lowered or raised by g, but what does that mean. The indices on R and S are more reasonable since they are directly related to differentials. In the curvilinear application, you might say at least x is in a Cartesian x-space, so at least xi and xi make sense. But what then do you say about x'i and x'i ? I guess you would look at x'i = Fi(x1, x2....) For a non-linear F, you could say that x'i is just an N-tuple. Suppose you come up with an arbitrary set of N functions and you name them Fi as shown here. Then x'i at least is well defined. You then get to this point: dx'i = Σk (∂Fi(x)/∂xk) dxk Rik = (∂Fi(x)/∂xk) But ∂Fi really is a contravariant vector! Once you have this R, you have g', and then you can say Fi(x1, x2....) ≡ g'ij(x'(x)) Fi(x1, x2....) and then Fi is fully defined. Somehow I could discuss this subject in my Review 1 or Review 2 sections. For non-linear F you say" x'i is an N-tuple that has an upper (contravariant) index but x'i is not a contravariant vector. If is only contravariant in the sense that dx'i is a contravariant vector. Even though x'i is just an N-tuple with an upper index, we can still define from it x'i = g'ijx'j and then x'i is different N-tuple vector which happens to have a lower index. ________________________________