## 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.

See the post ‘stokes theorem in geometric algebra‘ [PDF], where this topic has been revisited with this in mind.

# Original Post:

[Click here for a PDF of this post with nicer formatting]

In [3] 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.

# References

[1] L.D. Landau and E.M. Lifshits. *The classical theory of fields*. Butterworth-Heinemann, 1980.

[2] C. Doran and A.N. Lasenby. *Geometric algebra for physicists*. Cambridge University Press New York, Cambridge, UK, 1st edition, 2003.

[3] Peeter Joot. Stokes theorem derivation without tensor expansion of the blade [online]. http://sites.google.com/site/peeterjoot/math2009/stokesNoTensor.pdf.

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