Directed Integration and the Fundamental Theorem of Calculus

[kryptyk]

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:

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

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

$$\int_S d\omega = \oint_{\partial S} \omega$$

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

Any smooth manifold will have a pseudoscalar field at every point $$x$$ which we denote by $$I(x)$$. 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, $$I$$ is constant. In fact, one could use this as the defining condition for flatness. So then:

$$d\omega = \mid d\omega\mid I$$

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

 

[HallsofIvy]

It's very general and very abstract. You'd better have a good grounding in differential geometry and manifolds before you go that route!
For example
$$\int_S d\omega = \oint_{\partial S} \omega$$
is only true if the fundamental homology group of S is trivial!

 

[mathwonk]

I'm not sure what you are thinking of halls, but the theorem you have written above looks like stokes theorem, which I think does not have any hypotheses on homology?

 

[kryptyk]

The Need for Trivial Homology

I do think Halls is right.

If $$S$$ contains one or more internal points where field $$F$$ is singular then it becomes impossible to continuously shrink $$\partial S$$ to an arbitrarily small neighborhood around any internal point in $$S$$.

 

[kryptyk]

Boundary Normal

I think a key idea here is that if $$S$$ is an m-dimensional manifold then $$\partial S$$ is a closed (m-1)-dimensional manifold. Given an m-blade field $$I_m$$ on $$S$$ and an (m-1)-blade field $$I_{m-1}$$ on $$\partial S$$ each representing the tangent spaces of $$S$$ and $$\partial S$$ respectively, it is possible to find a normal vector field $$\mathbf{n}$$ on $$\partial S$$ such that $$I_{m} = I_{m-1} \wedge \mathbf{n}$$.

But then $$\mathbf{n}$$ is everywhere orthogonal to $$I_{m-1}$$ and so $$I_{m-1} \cdot\mathbf{n}$$ vanishes. Thus we have:

$$I_{m} = I_{m-1} \mathbf{n}$$

So we find an expression that relates the tangent spaces of $$S$$ and $$\partial S$$ at each point on $$\partial S$$.

 

[kryptyk]

Directed Measures

The following is a brief summary of some core ideas in Hestenes[98].

Let $$S$$ be parametrized by m coordinates $$\{x^1, x^2, \ldots, x^m\}$$.

Then at each point $$\mathbf{x}$$ in $$S$$ we have a covariant frame $$\{\mathbf{e}_k\}$$ given by:

$$\mathbf{e}_k = \frac{\partial\mathbf{x}}{\partial x^k}$$

It is now possible to define an m-blade (pseudoscalar) field $$I_m$$ giving the orientation of $$S$$:

$$I_m = \mathbf{e}_1 \wedge\mathbf{e}_2 \wedge \ldots \wedge\mathbf{e}_m$$

We then define the vector differentials:

$$d\mathbf{x}^k = d x^k \mathbf{e}_k$$

Now we can explicitly construct a directed measure on $$S$$:

$$d^m \mathbf{x} = d\mathbf{x}^1 \wedge d\mathbf{x}^2 \wedge \ldots \wedge d\mathbf{x}^m$$

But $$d x^k$$ are scalars, so we have:

$$d^m \mathbf{x} = d\mathbf{x}^1 \wedge d\mathbf{x}^2 \wedge \ldots \wedge d\mathbf{x}^m = \mathbf{e}_1 \wedge\mathbf{e}_2 \wedge \ldots \wedge\mathbf{e}_m d x^1 d x^2 \ldots d x^m = I_m d x^1 d x^2 \ldots d x^m = I_m d^m x$$

Now we can integrate a multivector field $$F$$ over $$S$$ using scalar calculus methods (i.e. Riemann integrals, iterated integrals):

$$\int_S d^m \mathbf{x} F = \int_S I_m d^m x F$$

For flat manifolds, $$I_m$$ is constant and we can pull it out of the integral to get

$$\int_S d^m \mathbf{x} F = I_m \int_S d^m x F$$

 

[HallsofIvy]

I'm not sure what you are thinking of halls, but the theorem you have written above looks like stokes theorem, which I think does not have any hypotheses on homology?

It has hypotheses on being "simply connected" which is equivalent to the first homology group being trivial.

 

[kryptyk]

Intuitive argument for the homology condition

Halls is right - but beyond the axiomatic "reasons" for this condition being necessary, it is possible to develop a strong intuitive argument just by looking at the properties of $$\partial S$$ as we shrink it around the neighborhood of a point in $$S$$. If a field $$F$$ defined on $$S$$ is not smooth as $$\partial S$$ shrinks around some point $$\mathbf x$$ then it is clear that the theorem must fail for certain boundaries since that would imply that

$$\int_{\partial S} F$$

would not be smooth as $$\partial S$$ is allowed to change smoothly through these points.

But,

$$\int_S d F$$

must change smoothly as $$\partial S$$ is allowed to change smoothly so that both integrals are well-defined and converge.

However, if the points of discontinuity are removable, we can still define a Lebesgue integral even if we cannot define a Riemann integral - and so the theorem would hold for all $$\partial S$$ that do not pass through one of these points.

 

[mathwonk]

is this some specuial version of stokes theorem i have not heard of, i.e. in "directed integration" as opposed to ordinary inmtegration of differential forms? certainly the usual one for integration of forms on a manifold with boundary has no hypotheses of simple connectivity at all, or am i losing my mind here?


see spivak, cal on manifolds, p.124, or guillemin and pollack differetial topology, p.183, or lang, analysis II, page 453.

perhaps hestenes version is something i am unfamiliar with?

or are you using forms that are not smooth?

for as smooth n-1 form w on a smootj n manifold M with boundary ?M, the integral of w over ?M equals the integral of dw over M. this is just the fundamental theorem of calculus, plus fubini's theorem, globalized via local ccordinate charts.

but surely you know all this, so I must be unaware of what type of situation you are speaking of.

what are your hypotheses?

 

[kryptyk]

Nevermind...

You're right, mathwonk. Simple connectivity is not required. Perhaps Halls was referring to contour integration using complex variables?

 

[laserblue]

http://modelingnts.la.asu.edu/pdf/DIF_FORM.pdf
http://modelingnts.la.asu.edu/pdf/UnifiedLang.pdf
http://modelingnts.la.asu.edu/html/CA_to_GC.html
Clifford Algebra to Geometric Calculus (1986)

Chapter 7 / Directed Integration Theory
7-1. Directed Integrals
7-2. Derivatives from Integrals
7-3. The Fundamental Theorem of Calculus
7-4. Antiderivatives, Analytic Functions and Complex Variables
7-5. Changing Integration Variables
7-6. Inverse and Implicit Functions
7-7. Winding Numbers
7-8. The Gauss-Bonnet Theorem