Peeter Joot's (OLD) Blog.

Math, physics, perl, and programming obscurity.

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 k-1 grade blade, the final result of that work was

\begin{aligned}\int( \nabla \wedge F ) \cdot d^k x =\frac{1}{{(k-1)!}} \epsilon^{ r s \cdots t u } \int da_u \frac{\partial {F}}{\partial {a_{u}}} \cdot (dx_r \wedge dx_s \wedge \cdots \wedge dx_t)\end{aligned} \hspace{\stretch{1}}(7.44)

Let’s expand this in coordinates to attempt to get the equivalent expression for an antisymmetric tensor of rank k-1.

Starting with the RHS of 7.44 we have

\begin{aligned}F &= \frac{1}{{(k-1)!}}F_{\mu_1 \mu_2 \cdots \mu_{k-1} }\gamma^{\mu_1} \wedge \gamma^{ \mu_2 } \wedge \cdots \wedge \gamma^{\mu_{k-1}} \\ dx_r \wedge dx_s \wedge \cdots \wedge dx_t &=\frac{\partial {x^{\nu_1}}}{\partial {a_r}}\frac{\partial {x^{\nu_2}}}{\partial {a_s}}\cdots\frac{\partial {x^{\nu_{k-1}}}}{\partial {a_t}}\gamma_{\nu_1} \wedge \gamma_{ \nu_2 } \wedge \cdots \wedge \gamma_{\nu_{k-1}}da_r da_s \cdots da_t\end{aligned} \hspace{\stretch{1}}(7.45)

We need to expand the dot product of the wedges, for which we have

\begin{aligned}\left( \gamma^{\mu_1} \wedge \gamma^{ \mu_2 } \wedge \cdots \wedge \gamma^{\mu_{k-1}} \right) \cdot\left( \gamma_{\nu_1} \wedge \gamma_{ \nu_2 } \wedge \cdots \wedge \gamma_{\nu_{k-1}}\right) ={\delta^{\mu_{k-1}}}_{\nu_1} {\delta^{ \mu_{k-2} }}_{\nu_2} \cdots {\delta^{\mu_{1}} }_{\nu_{k-1}}\epsilon^{\nu_1 \nu_2 \cdots \nu_{k-1}}\end{aligned} \hspace{\stretch{1}}(7.47)

Putting all the LHS bits together we have

\begin{aligned}&\frac{1}{{((k-1)!)^2}} \epsilon^{ r s \cdots t u } \int da_u \frac{\partial {}}{\partial {a_{u}}} F_{\mu_1 \mu_2 \cdots \mu_{k-1} }{\delta^{\mu_{k-1}}}_{\nu_1} {\delta^{ \mu_{k-2} }}_{\nu_2} \cdots {\delta^{\mu_{1}} }_{\nu_{k-1}}\epsilon^{\nu_1 \nu_2 \cdots \nu_{k-1}}\frac{\partial {x^{\nu_1}}}{\partial {a_r}}\frac{\partial {x^{\nu_2}}}{\partial {a_s}}\cdots\frac{\partial {x^{\nu_{k-1}}}}{\partial {a_t}}da_r da_s \cdots da_t \\ &=\frac{1}{{((k-1)!)^2}} \epsilon^{ r s \cdots t u } \int da_u \frac{\partial {}}{\partial {a_{u}}} F_{\mu_1 \mu_2 \cdots \mu_{k-1} }\epsilon^{\mu_{k-1} \mu_{k-2} \cdots \mu_{1}}\frac{\partial {x^{\mu_{k-1}}}}{\partial {a_r}}\frac{\partial {x^{\mu_{k-2}}}}{\partial {a_s}}\cdots\frac{\partial {x^{\mu_1}}}{\partial {a_t}}da_r da_s \cdots da_t \\ &=\frac{1}{{((k-1)!)^2}} \epsilon^{ r s \cdots t u } \int da_u \frac{\partial {}}{\partial {a_{u}}} F_{\mu_1 \mu_2 \cdots \mu_{k-1} }{\left\lvert{\frac{\partial(x^{\mu_{k-1}},x^{\mu_{k-2}},\cdots,x^{\mu_1})}{\partial(a_r, a_s, \cdots, a_t)}}\right\rvert}da_r da_s \cdots da_t \\ \end{aligned}

Now, for the LHS of 7.44 we have

\begin{aligned}\nabla \wedge F &=\gamma^\mu \wedge \partial_\mu F \\ &=\frac{1}{{(k-1)!}}\frac{\partial {}}{\partial {x^{\mu_k}}} F_{\mu_1 \mu_2 \cdots \mu_{k-1}}\gamma^{\mu_k} \wedge\gamma^{\mu_1} \wedge \gamma^{ \mu_2 } \wedge \cdots \wedge \gamma^{\mu_{k-1}} \end{aligned}

and the volume element of

\begin{aligned}d^k x &=\frac{\partial {x^{\nu_1}}}{\partial {a_1}}\frac{\partial {x^{\nu_2}}}{\partial {a_2}}\cdots\frac{\partial {x^{\nu_{k}}}}{\partial {a_k}}\gamma_{\nu_1} \wedge \gamma_{ \nu_2 } \wedge \cdots \wedge \gamma_{\nu_k}da_1 da_2 \cdots da_k\end{aligned}

Our dot product is

\begin{aligned}\left(\gamma^{\mu_k} \wedge\gamma^{\mu_1} \wedge \gamma^{ \mu_2 } \wedge \cdots \wedge \gamma^{\mu_{k-1}} \right) \cdot\left( \gamma_{\nu_1} \wedge \gamma_{ \nu_2 } \wedge \cdots \wedge \gamma_{\nu_k} \right)={\delta^{\mu_{k-1}}}_{\nu_1} {\delta^{ \mu_{k-2} }}_{\nu_2} \cdots {\delta^{\mu_{1}} }_{\nu_{k-1}}{\delta^{\mu_{k}} }_{\nu_{k}}\epsilon^{\nu_1 \nu_2 \cdots \nu_{k}}\end{aligned} \hspace{\stretch{1}}(7.48)

The LHS of our k-form now evaluates to

\begin{aligned}(\gamma^\mu \wedge \partial_\mu F) \cdot d^k x &= \frac{1}{(k-1)!}\frac{\partial }{\partial {x^{\mu_k}}} F_{\mu_1 \mu_2 \cdots \mu_{k-1}}{\delta^{\mu_{k-1}}}_{\nu_1} {\delta^{ \mu_{k-2} }}_{\nu_2} \cdots {\delta^{\mu_1} }_{\nu_{k-1}}{\delta^{\mu_k} }_{\nu_k}\epsilon^{\nu_1 \nu_2 \cdots \nu_k}\frac{\partial {x^{\nu_1}}}{\partial {a_1}}\frac{\partial {x^{\nu_2}}}{\partial {a_2}} \cdots \frac{\partial {x^{\nu_k}}}{\partial {a_k}} da_1 da_2 \cdots da_k \\ &= \frac{1}{(k-1)!}\frac{\partial }{\partial {x^{\mu_k}}} F_{\mu_1 \mu_2 \cdots \mu_{k-1}}\epsilon^{\mu_{k-1} \mu_{k-2} \cdots \mu_1 \mu_k}\frac{\partial {x^{\mu_{k-1}}}}{\partial {a_1}}\frac{\partial {x^{\mu_{k-2}}}}{\partial {a_2}} \cdots \frac{\partial {x^{\mu_1}}}{\partial {a_{k-1}}}\frac{\partial {x^{\mu_k}}}{\partial {a_k}} da_1 da_2 \cdots da_k \\ &= \frac{1}{(k-1)!}\frac{\partial }{\partial {x^{\mu_k}}} F_{\mu_1 \mu_2 \cdots \mu_{k-1}}{\left\lvert{\frac{\partial(x^{\mu_{k-1}},x^{\mu_{k-2}},\cdots x^{\mu_1},x^{\mu_k})}{\partial(a_1, a_2, \cdots, a_{k-1}, a_k)}}\right\rvert} da_1 da_2 \cdots da_k \\ \end{aligned}

Presuming no mistakes were made anywhere along the way (including in the original Geometric Algebra expression), we have arrived at Stokes Theorem for rank k-1 antisymmetric tensors F

\boxed{ \begin{aligned}&\int\frac{\partial }{\partial {x^{\mu_k}}} F_{\mu_1 \mu_2 \cdots \mu_{k-1}}{\left\lvert{\frac{\partial(x^{\mu_{k-1}},x^{\mu_{k-2}},\cdots x^{\mu_1},x^{\mu_k})}{\partial(a_1, a_2, \cdots, a_{k-1}, a_k)}}\right\rvert} da_1 da_2 \cdots da_k \\ &= \frac{1}{(k-1)!} \epsilon^{ r s \cdots t u } \int da_u \frac{\partial }{\partial {a_u}} F_{\nu_1 \nu_2 \cdots \nu_{k-1} }{\left\lvert{\frac{\partial(x^{\nu_{k-1}},x^{\nu_{k-2}}, \cdots ,x^{\nu_1})}{\partial(a_r, a_s, \cdots, a_t)}}\right\rvert} da_r da_s \cdots da_t \end{aligned} } \hspace{\stretch{1}}(7.49)

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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