Pushforward - differential of a mapping

Pushforward[expr, mapId]
allows one to calculate the pushforward - mapping differential of a tensor field
  • mapId - variable - the mapping identifier i.e. mapId : dom1 ---> dom2.
  • expr - expression - a tensor expression.
  • The mapping differential is a linearly defined operation on [k,0] tensors at a point only. The definition is as follows.
  • Let M and N be manifolds of dimensions m=dim(M), n=dim(N). Let F be mapping between the manifolds: F:MN defined by functions:
  • where {x1, x2,..xm} are local coordinates on M and {y1, y2,..yn} are local coordinates on N (in some domains).
  • The differential of F is the mapping F`*` of corresponding tangent spaces F`*`:TMTN at a point.
  • - For any [1,0] tensor field T on M F`*`(T) is tensor field =Pushforward[T, F] on N with components in local coordinates.
  • The formulas considered above completely define the linear restriction operator Pushforward.
The following example shows how the operator can be used.
Let M be 2-dimentional sphere S2 and N be 3-dimensional Euclidean space R3. Let F:MN be standard embedding of sphere S2 into R3.
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Declare 1-forms ej and uk for corresponding coframes:
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Declare vectors for corresponding frames:
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Declare Euclidean space - R3:
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Declare coframe on R3:
Declare frame on R3:
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Declare metric on R3 (standard flat metric):
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Declare sphere - S2:
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Declare coframe on S2:
Declare frame on S2:
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Declare definite mapping F: S2R3:
Visualize the mapping:
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Verify that we are on the sphere:
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Calculate metric induced on the sphere using Pullback operator:
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One can calculate restriction of any [k,0] tensor field on S2 under the mapping: For of frame vectors:
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Restriction of tensor product CircleTimes[d[x], d[z]]:
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For metric tensor:
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Declare abstract mapping between S2 and R3:
For abstract mapping :
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Who is who?
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