atlas 2 for Mathematica®

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Hodge - Hodge operator

Hodge[expr]
allows one to calculate Hodge operator on an expression containing p-forms
  • expr - any expression containing p-forms
  • If a metric is presented then the Hodge operator is defined completely by the following:
  • - Let pM be vector bundle of p-forms on manifold M of dimension n=dim(M) and metric g.
  • - For any integer 0 < p ≤ n let us define Hodge operator * as such unique isomorphism of vector bundles * : pM ---> n-pM which has the following property.
  • - For any , which belong to p we have (*())=g(, )g where g is volume form on M induced by metric g.
  • Let s be the number of -1 in the signature of metric g (in the ATLAS package the integer is represented by sgn variable) then the following equations take place:
  • - *1=g and *g=1
  • - (*)2=-1p(n-p)+s on vector bundle pM.