Coframe - declaration of coframe 1-forms

Coframe[idj, {j, 1, n}]
sets idj as coframe 1-forms e.g. Coframe[ej, {j, 1, n}] - sets 1-forms e1, e2, ..en as a coframe, here n is dimension of working manifold i.e. dim= n (see Dimension); n can be symbolic or integer. This syntax must be used if a coframe defined before frame (see Frame).
Coframe[idjexprj, {j, 1, n}]
sets idj as coframe 1-forms so that idj=fj where fj is some 1-form or liner expression on 1-forms (see examples below), here n is dimension of working manifold i.e. dim= n; n can be symbolic or integer. This syntax must be used if a coframe defined before frame (see Frame).
Coframe[id1expr1, id2expr2, ...idnexprn]
sets id1, id2, ..idn as coframe 1-forms e.g. Coframe[e1d[x], e2d[y]] sets 1-forms e1, e2 as a coframe which is d[x], d[y] where x, y are 0-forms. This syntax must be used if a coframe defined before frame (see Frame).
Coframe[idj]
sets idj as coframe 1-forms e.g. Coframe[ej] - sets 1-forms e1, e2, ..en as a coframe, here n is dimension of working manifold i.e. dim= n (see Dimension); n can be symbolic or integer. The dimension is set by the Frame declaration. This syntax must be used if a coframe defined after frame (see Frame).
  • The Coframe procedure allows one to declare coframe 1-forms. If the third type of calling sequence is used, then the local coordinates, coframe 1-forms and dimension of the working manifold have been defined automatically.
  • id - identifier for indexed variable - the coframe 1-forms
  • n - dimension of working manifold (a variable or integer)
  • idiexpri - equation where idi is indexed variable - coframe 1-form and expri is decomposition of the 1-form on exact 1-forms.
  • Warning! Only indexed variables can be used as coframe 1-forms. The name of the indexed variable must be one and the same for all coframe 1-forms e.g. e1, e2, ..en or 1, 2, ..k and so on.
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Declare forms:
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Declare vectors:
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Declare functions:
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Declare coframe with 1-forms: e1, e2, ..en (dim = n):
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Function F is declared as follows F=F(zk); now k=1, 2, ..n:
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For function f we have:
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Declare frame vectors (see Frame):
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"To basis" decomposition:
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Interior product of vector Y and 1-form e[j]:
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"To basis" decomposition:
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Declare vectors:
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Declare functions:
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Declare coframe with 1-forms: e1, e2, e3 (dim = 3):
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Functions F, z1 are declared as follows F=F(zk) and z1=z1(x, y):
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Declare frame vectors (see Frame):
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"To basis" decomposition:
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Interior product of vector Z and 1-form e[j]:
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"To basis" decomposition:
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Declare functions:
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Declare constant :
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Declare coframe:
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Declare frame:
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Declare metric (see Metric):
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Simple calculation:
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Declare constant rg:
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Declare vectors:
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Declare forms:
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Declare coframe:
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Declare frame:
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Declare metric:
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Simple calculations:
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Declare vectors:
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Declare functions:
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Declare coframe:
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Declare frame:
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Simple calculations:
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Declare constant :
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Declare forms:
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Declare vectors:
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Declare functions:
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Declare coframe:
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Declare frame:
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Simple calculations:
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