Atlas package examples

Description:

• To make calculation in modern differential geometry  it is necessary to define manifold , coframe  1-forms, frame  vectors etc. Thus the following sequence of steps is natural for any such a calculation:
  1. Name the manifold  (or its domain ) you are working at ( atlas[Domain] ).
  2. Declare constants, vectors, p-forms, tensors  and functions  ( atlas[Constants] ,
       atlas[Vectors] , atlas[Forms] , atlas[Tensors] , atlas[Functions] ).
  3. Declare coframe  1-forms ( atlas[Coframe] ).
  4. Declare frame  vector fields ( atlas[Frame] ).
  5. Declare metric  if needed ( atlas[Metric] ).
  6. Calculate connection  and curvature  if needed ( atlas[Connection] , atlas[Curvature] ).
  7. Make any necessary calculations.
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  1. Name other manifold if needed.
  2. Declare coframe  1-forms for the manifold.
  3. Declare mappings  between manifolds if needed ( atlas[Mapping] ). .
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• User may declare any number of manifolds  (or its domains ) in one session. Each manifold can have its own dimension , coframe, frame, metric, connection  etc. Moreover, any number (and kind) of mappings  between manifolds can be declared.

Examples:

Coordinate system changing
Plane curves
Surface geometry
Winding line on a torus
"Abstract" calculations  
Define a manifold as a whole (3-sphere)
Ricci - flat warped product
S[1] - fibration  

See Also:

atlas[types] , atlas[simp] , atlas .