old duality notes

old duality notes (keep, these notes were replaced in tensor doc on 4.11.12) 1. The reciprocal base vectors En are more usually defined as being those vectors which satisfy the equations En em = δn,m where the em are known. Each En vector has N components so the full set of En vectors has N2 components. As n and m take all values, En em = δn,m represents N2 linear equations. A solution exists since S is invertible (the em form a complete set). The solution is unique and in fact gives our assumed definition above En ≡ g'ni ei . Here is a fast solution of this Cramer's Rule problem using matrix notation: En em = δn,m => ij(En)i(em)j = δn,m => (En)i ij Sjm = δn,m Let Ani ≡ (En)i . Then have A S = 1, => A = ( S)-1 = S-1 -1 = R g = g' ST (end Sec 5f). Therefore A = g' ST => (En)i = Ani = g'nj (ST)ji = g'nj (ej)i => En = g'nj ej . QED 2. In general, if one has Bn bm = δn,m, the vectors Bm are said to be "reciprocal" to the bm and vice versa, so the vectors En are reciprocal to the tangent base vectors en. 3. Some authors refer to Bn bm = δn,m as a "duality relation" and either set of vectors is "dual to" the other set. The En are referred to as the dual vectors to en. 4. If the bm are true tensorial vectors, then the Bn will be as well and then Bn bm is a tensorial scalar. Therefore if Bn bm = δn,m, then so also B'n b'm = δn,m in x'-space, where bm' = Rbm and B'n = RBn. For example, E'n e'm = δn,m in x'-space where em' = Rem and E'n = REn . 5. In section (e) we shall encounter another dual pair Un um = U'n u'm = δn,m which is associated with the inverse transformation x = F-1(x'). 6. One major significance of the equation Bn bm = δn,m is that it allows the following expansions: V = Σn kn Bn where km = V bm V = Σn cn bn where cm = V Bm so that for example bm V = bm [Σn kn Bn] = Σn kn bm Bn = Σn kn δm,n = km. These expansions are explored in section (f) below for the two dual sets En, en and Un, un.