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
Connection with torsion
S[1] - fibration  

See Also:  

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