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Mapping - declaration of a mapping between manifolds (domains)

Mapping[f, M, N]
Mapping[f, M, N, y1→f1(x1, x2...xm), y2→f2(x1, x2...xm), ...,
yn→fn(x1, x2...xm)]
allows one to declare a mapping between manifolds (or its domains)
  • f - variable - the mapping identifier i.e. f : M N M - variable - first domain identifier N - variable - second domain identifier
  • Once a mapping is defined, the user can calculate the pullback of any [0,k] tensor field under the mapping (see Pullback).
  • The Mapping procedure can be used in two ways:
  • - Mapping(f, M, N) - declares an abstract mapping between manifolds or domains such that: F:MN
  • - Mapping[f,M,N,y1→f1(x1,x2...xm),y2→f2(x1,x2...xm),..., yn→fn(x1,x2...xm)] - declares a mapping between manifolds such that: F:MN. The mapping defined by functions:
  • where m=dim(M), n=dim(N); {x1,x2,..xm} are local coordinates on M and {y1,y2,..yn} are local coordinates on N.
EXAMPLESCLOSE ALL
Basic Examples   (1)
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Declare 1-forms ej and uk for corresponding coframes:
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Declare vectors:
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Declare domain S2 (sphere):
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Declare coframe on S2:
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Declare frame :
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Declare domain R2(plane):
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Declare coframe on R2:
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Frame declaration:
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Declare abstract mapping between the sphere and the plane:
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Declare definite mapping between the sphere and the plane:
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Visualize the mapping:
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Declare another definite mapping between the sphere and the plane:
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Visualize the mapping:
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Clarify "who is who".
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One can calculate pullback of any [0,k] tensor field under a mapping (see Pullback):
pullback of exterior product d()d() under mapping :
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pullback of "rotation" 1-form under mapping psi:
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pullback of "rotation" 1-form under mapping :
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pullback of exterior product Wedge[d[x], d[y]] under abstract mapping :
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