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Working scratch document, dated 1.11.15 by Phil's initials, in the folder of files for the May 2015 update of his curvilinear coordinates tensor document. It lists basis vector, scale factor and metric relations (g, R, S, e_n, u_n, E_n) with equation numbers, then applies substitution rules such as g' to g and R to S, with the changed results to be marked in red. The extracted equations are partly garbled, so the details are approximate.
AI-written summary; may contain errors. This description is approximate.
Extracted text (machine-read; may contain errors)
This is the Title PhL 1.11.15
Note that page numbering is turned on in this template and view is 125%,
located in phil/roaming/microsoft/templates size about 219K.
('n)i = 'ij (u'n)j ('n)i = 'ij (U'n)j [ 'n = ' u'n 'n = ' U'n]
(un)i = Rij(u'n)j (Un)i = Rij(Un')j [ un = S u'n Un = S Un' ]
(n)i = Sji('n)j (n)i = Rji('n)j [ n = RT 'n n = RT 'n ]
(u'n)j = Rjn (U'n)i = g'ijSnj = gnjRij (un)i = δi,n (Un)i = gni
('n)i = 'ijRjn = Sjijn ('n)i = Sni (n)i = gni (n)i = δn,i
u'n u'm = nm => |u'n| = = hn (scale factor) U'n = gni u'i
U'n u'm = δn,m u'n = ni U'i
U'n U'm = gnm => |U'n| =
un um = nm => |un| = = hn (scale factor) Un = gni ui
Un un = δn,m un = ni Ui
Un Um = gnm => |Un| =
('n)a(u'n)b = δa,b or Σn 'n u'nT = 1 (6.5.3)
***************************
(en)i = gij (en)j (en)i = gij (en)j (5.8.4)
(e'n)i = Rij(en)j (e'n)i = Rij(en)j
(e'n)i = Sji(en)j = Rij(en)j (e'n)i = Sji(en)j = Rij(en)j (2.5.1)
(en)i = Sin = Rni (en)i = gijRnj = g'njSij (en')i = δni (e'n)i = g'ni
(en)i = gijSjn = Rjig'jn (en)i = Rni (en')i = g'ni (e'n)i = δni
(6.3.3) (6.3.10)
en em = g'nm => |en| = = h'n (scale factor) en = g'ni ei
en em = δnm en = g'ni ei
en em = g'nm => |en| = . (6.2.4)
e'n e'm = g'nm => |e'n| = = h'n (scale factor) e'n = g'ni e'i
e'n e'm = δnm e'n = g'ni e'i
e'n e'm = g'nm => |e'n| = . (6.2.7)
(en)i(en)j = δij (6.2.16) (7.18.1)
Translate the above with these rules
g'↔ g R ↔ S en → u'n e'n → un En → U'n E'n → Un (6.5.2)
g'↔ g R ↔ S en → u'n e'n → un en → u'n e'n → un (6.5.2)
Now apply these rules one at a time and make red after:
(u'n)i = g'ij (u'n)j (u'n)i = g'ij (u'n)j (5.8.4)
(un)i = Sij(u'n)j (un)i = Sij(u'n)j
(un)i = Rji(u'n)j = Sij(u'n)j (un)i = Rji(u'n)j = Sij(u'n)j (2.5.1)
(u'n)i = Rin = Sni (u'n)i = g'ijSnj = gnjRij (un)i = δni (un)i = gni
(u'n)i = g'ijRjn = Sjigjn (u'n)i = Sni (un)i = gni (un)i = δni
(6.3.3) (6.3.10)
u'n u'm = gnm => |u'n| = = hn (scale factor) u'n = gni u'i
u'n u'm = δnm u'n = gni u'i
u'n u'm = gnm => |u'n| = . (6.2.4)
un um = gnm => |un| = = hn (scale factor) un = gni ui
un um = δnm un = gni un
un um = gnm => |un| = . (6.2.7)
(u'n)i(u'n)j = δij (6.2.16) (7.18.1)