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Frame - declaration of frame vectors

Frame[idj, {j, 1, n}]
sets idj as frame vectors e.g. Frame[Ej, {j, 1, n}] - sets vectors 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 frame defined before coframe (see Coframe).
Frame[idj→ exprj, {j, 1, n}]
sets idj as frame vectors so that idj=fj where fj is some vector or liner expression on vectors (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 frame defined before coframe (see Coframe).
Frame[id1→ expr1, id2→ expr2, ...idn→ exprn]
sets id1, id2, ..idn as frame vectors e.g. Frame[E1→ x, E2→ y] sets vectors E1, E2 as a frame which is x, y where x, y are 0-forms. This syntax must be used if a frame defined before coframe (see Coframe).
Frame[idj]
sets idj as frame vectors e.g. Frame[Ej] - sets vectors 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 Coframe declaration. This syntax must be used if a frame defined after coframe (see Coframe).
  • The Frame procedure allows one to declare frame vectors. If the third type of calling sequence is used, then the local coordinates, frame vectors and dimension of the working manifold have been defined automatically.
  • id - identifier for indexed variable - the frame vectors
  • n - dimension of working manifold (a variable or integer)
  • idi→ expri - equation where idi is indexed variable - frame vector and expri is decomposition of the vectors on "natural" vectors (see Nat).
  • Warning! Only indexed variables can be used as frame vectors. The name of the indexed variable must be one and the same for all frame vectors e.g. E1, E2, ..En or 1, 2, ..k and so on.
EXAMPLESCLOSE ALL
Basic Examples   (6)
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Declare forms:
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Declare vectors:
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Declare functions:
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Declare frame with vectors: 1, 2, ..n (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 coframe 1-forms (see Coframe):
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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 frame with vectors: 1, 2, 3 (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 coframe 1-forms (see Coframe):
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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 vectors:
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Declare functions:
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Declare constant :
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Declare frame:
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Declare coframe:
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Declare metric (see Metric):
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Simple calculation:
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Declare functions:
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Declare constant rg:
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Declare vectors:
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Declare forms:
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Declare frame:
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Declare coframe:
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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 frame:
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Declare coframe:
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Simple calculations:
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Declare forms:
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Declare vectors:
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Declare functions:
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Declare frame:
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Declare coframe:
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Simple calculations:
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