I'm still new to much of this stuff, so I do not claim to be an expert. But I thought I'd still comment. I'm trying to better understand the geometric calculus of David Hestenes:(adsbygoogle = window.adsbygoogle || []).push({});

http://modelingnts.la.asu.edu/

A fairly common form for the Fundamental Theorem of Calculus is:

[tex]\int_S d\omega = \oint_{\partial S} \omega[/tex]

Typically only scalar [tex]\omega[/tex] are considered. However, it is also possible to consider multivector-valued [tex]\omega[/tex] in which case [tex]d\omega[/tex] is a directed measure rather than a scalar.

Any smooth manifold will have a pseudoscalar field at every point [tex]x[/tex] which we denote by [tex]I(x)[/tex]. For an n-dimensional manifold, this will be an n-blade which identifies the tangent space at the point x. I talked a bit about blades in this thread:

https://www.physicsforums.com/showthread.php?t=114604

For flat manifolds, [tex]I[/tex] is constant. In fact, one could use this as the defining condition for flatness. So then:

[tex]d\omega = \mid d\omega\mid I[/tex]

There seem to be some advantages to this approach. Any ideas?

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# Directed Integration and the Fundamental Theorem of Calculus

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