Stokes Theorem for antisymmetric tensors.
Posted by peeterjoot on January 18, 2011
Obsolete with potential errors.
This post may be in error. I wrote this before understanding that the gradient used in Stokes Theorem must be projected onto the tangent space of the parameterized surface, as detailed in Alan MacDonald’s Vector and Geometric Calculus.
In  I worked through the Geometric Algebra expression for Stokes Theorem. For a grade blade, the final result of that work was
Let’s expand this in coordinates to attempt to get the equivalent expression for an antisymmetric tensor of rank .
Starting with the RHS of 7.44 we have
We need to expand the dot product of the wedges, for which we have
Putting all the LHS bits together we have
Now, for the LHS of 7.44 we have
and the volume element of
Our dot product is
The LHS of our k-form now evaluates to
Presuming no mistakes were made anywhere along the way (including in the original Geometric Algebra expression), we have arrived at Stokes Theorem for rank antisymmetric tensors
The next task is to validate this, expanding it out for some specific ranks and hypervolume element types, and to compare the results with the familiar 3d expressions.
 L.D. Landau and E.M. Lifshits. The classical theory of fields. Butterworth-Heinemann, 1980.
 C. Doran and A.N. Lasenby. Geometric algebra for physicists. Cambridge University Press New York, Cambridge, UK, 1st edition, 2003.
 Peeter Joot. Stokes theorem derivation without tensor expansion of the blade [online]. http://sites.google.com/site/peeterjoot/math2009/stokesNoTensor.pdf.