archive old sections 1_2 and 1_3

archive old Section 1.2 and 1.3 PhL 2.23.17 1.2 Expansions of a vector and use of primes and parentheses Note: We write (a)i as a component of vector a , but (en)i as a component of en . In the first case the a in (a)i is not bolded, but since en is decorated with a label n, it gets bolded. It is just our convention. Any vector a can be expanded on either set of basis vectors ei (Frame S) or e'i (Frame S') so that, with implied summation on i, a = ai ei = a'i e'i . ai = a ei a'i = a e'i . (1.2.1) If some other vector named a' is lurking in the wings, one might want to be more careful labeling components. A safe method is this: a = (a)i ei = (a)'i e'i (a)i = a ei (a)'i = a e'i a' = (a')i ei = (a')'i e'i (a')i = a' ei (a')'i = a' e'i . (1.2.2) Here, a prime inside a parentheses is part of the vector name, whereas a prime outside a parentheses denotes a vector component in Frame S' (whereas no prime outside means a component in Frame S). Unless the relationship between vectors a and a' has a certain simple form, it is very likely that (a')i ≠ (a)'i . In this case the notation a'i would be ambiguous since one doesn't know whether it refers to (a')i or (a)'i. It is true that the notation ai could be unambiguously identified with (a)i, but we shall maintain the parentheses just to be uniform. Matrix Notation to show how components are related. Let R be the transformation appearing in the Basis Theorem (1.1.29) such that e'n = Rnm em and en = Re'n. Consider the following expansion of vector a on the basis vectors e'j, a = (a)'j e'j // (1.2.2) = (a)'j { Rji ei } // e'n = Rnm em , linear combination of vectors = (a)'j { (R-1)ij ei } // R = (R-1)T real orthogonal rotation = { (R-1)ij(a)'j )ei . // reorder (1.2.3) Comparing this to a = (a)i ei of (1.2.2) we conclude that, since ei is a complete basis, (a)i = (R-1)ij(a)'j so (a)'i = Rij(a)i (1.2.4) where R-1 is a 3x3 rotation matrix. We can repeat the above discussion replacing a with a' with this result, (a')i = (R-1)ij(a')'j . so (a')'i = R ij(a')i (1.2.5) These matrix equations are convenient for computing the components of a vector on the ei basis if they are known in the e'i basis (and vice versa) . Dot Products Consider two normal (normally transforming) vectors a' = Ra and b' = Rb, We know using (1.1.38) that the quantity a b = [Ra] [Rb] = a' b'. As an example, one then has a a = a' a' which says |a|2 = |a'|2. Thus a real orthogonal transformation R is one which preserves the length of a vector. Note that both regular rotations (detR=1) and reflections (detR=-1) have this property. Since the dot product a b has the same value in Frame S as in Frame S', it is a "rotational scalar", as distinct from a "scalar" which sometimes just means a 1-tuple. The basis vectors are not normal vectors because they are back rotated, meaning en' = R-1en. We can of course "front rotate" a basis vector to get qn = Ren but these qn are not the basis vectors en'. This means that the dot product a en is not a rotational scalar, even though it is a dot product of two "vectors". So one will not have a en = a' e'n. In fact a en = an a' e'n = [Ra] [R-1en] = [R2a] en = [R2a]n ≠ a en . // (1.1.38) On the other hand, we still have en em = [Re'n] [Re'n] = e'n e'm = δnm . Although this dot product is the same in both frames, it is not a rotational scalar because it is in fact a rank-2 tensor known as the metric tensor. The distinction is minor for rotations because in fact the dot product is the same number in both frames, either 0 or 1. 1.3 Special case where a'i is unambiguous We shall now examine the type of relationship between a' and a in which (a')i = (a)'i and therefore we can use the notation a'i without ambiguity. First, the components (a)'i and (a)i are related in the following simple manner, using (1.1.8), (a)'n = e'n a = (e'n)m (a)m = Rnm(a)m . (1.3.1) Now suppose we define a new vector a' in this way, a' ≡ Ra . (1.3.2) If a vector a transforms into a' according to (1.3.2), we say it is a "vector under rotations" which means it "transforms as a vector under rotations". When written in Frame S components this says (a')n = Rnm(a)m . (1.3.3) Comparison of (1.3.1) and (1.3.3) shows that (a)'n = (a')n (1.3.4) and therefore in this case we can use a'n ≡ (a)'n = (a')n . (1.3.5) Thus, if the vectors a and a' are related by a' = Ra where R is the rotation appearing in en = R e'n , then we can dispense with the parentheses as shown in (1.3.4). We still have (a')'i which requires parentheses. Example 1: Consider the equation in the Basis Theorem (1.1.30) , e'n = R-1 en . Since this is not of the form a' ≡ Ra, we may not dispense with the parentheses. In fact from (1.1.8) we have (en)'i = Rin (e'n)i = Rni (1.3.6) and these are not the same because rotation matrices are not symmetric. Unlike normal vectors for which one writes the transform a' = Ra, the basis vectors are "back-rotated" as noted earlier so e'n = (R-1)en. Active and Passive One can think of a' = Ra as an active rotation of vector a into another vector a' within Frame S. In this case, the components of a' are (a')i. The alternative is to think of vector a as not moving at all in Frame S, but the basis vectors are back-rotated from en to e'n taking us to Frame S'. In this back-rotated basis the components of a are (a)'i. This is the passive view of a rotation and is in fact the view we take in most of this document because we want to observe activities in Frame S from Frame S' and vice versa. Each view has its usefulness and we have just shown that if a' = Ra, then (a)'n = (a')n. In the active view, the "apparatus" is rotated and the axes stay put, while in the passive view the apparatus stays put but the axes are back-rotated. There is a third view in which the apparatus and the axes are both rotated in the same direction, and this view is useful in the discussion of covariance of equations (e.g. Lucht Tensor ***). One can regard the three views as three "experiments" one might perform. Example 2: Soon we shall be dealing with the Fig 1 equation r' = r - b. Since this is not of the form r' ≡ Rr , we may not dispense with the parentheses, and we expect that (r')i and (r)'i will be different. Footnote: More generally, if R is the linearized version of some general transformation x' = F(x) at a point x, so that dx' = R(x) dx, then (1.3.2) that a' = Ra says that a "transforms as a vector with respect to the underlying transformation F ". In general R(x) is a combination of rotation and stretch and is a function of location. In our current document we deal only with R(x) = R = a rotation that is the same at all points in space. It turns out however that the notation a'i is unambiguous in the general case as well as we shall now show. Lucht Tensor uses a different notation for basis vectors, and to make the connection between our current document and Tensor one must take {en,e'n} → {un,en} Frame S = {en} → Frame S = {un} en = R e'n → un = Rei Frame S' = {e'n} → Frame S' = {en} . Then in the language of Tensor where is the "covariant dot product", one has (implied summations), (a')n = a' un = (Ra) (Ren) = (Ra)i(Ren)i = Rij (a)j Rik (en)k = (RijRik) (a)j(en)k = δjk(a)j(en)k = (a)j(en)j = a en = (a)'n of this document. In this general case, within Frame S the un are still axis-aligned basis vectors but the en are generally not axis-aligned and are generally not unit vectors. For example, in spherical coordinates e1 = , e2 = r and e3 = rsinθ as shown in (E.6.8) below.