What Is the Purpose of Exterior Forms in Differential Geometry?

[phoenix95]

Hello there,
I had some questions regarding k-forms. I was looking in the wiki page of differential forms(https://en.wikipedia.org/wiki/Differential_form) and noticed that it was was introduced to perform integration independent of the co-ordinates. I am not clear how? Is this because given a function(say f) on the manifold, one can use the co-ordinate chart(Φ:M→ℝ^m) to define the same function in the euclidean space(as f(Φ^-1)) and integrate over there?

 

[Orodruin]

In general, a $p$-form is the natural description of a directed $p$-dimensional hypervolume element, it has all the properties you would expect, in particular complete antisymmetry in its arguments. As comparison, compare to the regular surface element $d\vec S$ that you most likely encountered when doing surface integrals in regular vector analysis. If you parametrise the integration surface by two parameters $s$ and $t$, it would be given by
$$
\newcommand{\dd}[2]{\frac{\partial #1}{\partial #2}}
d\vec S = \dd{\vec x}{s}\times \dd{\vec x}{t}\, ds\, dt,
$$
where the partial derivatives of $\vec x$ are the tangent vectors keeping the other parameter constant. In the same fashion, integrating a $p$-form $\omega$ over a $p$-dimensional hypersurface $H$ parametrised by $p$ parameters $t_p$ will give you an integral on the form
$$
\int_H \omega = \int_{H^*} \omega(\dot\gamma_1, \ldots, \dot\gamma_p) dt_1 \ldots dt_p,
$$
where $\dot\gamma_i$ is the tangent vector for the $t_i$ coordinate line and $H^*$ is the parameter range of the $t_i$.

Note that what you are actually integrating is the $p$-form itself, multiplying by some scalar function $f$ gives you a new $p$-form. In general, $p$ does not have to be equal to the dimension of the manifold, but it should be equal to the dimensions of the submanifold (or collection of submanifolds) that you integrate over. For example, $f\omega$ is a $p$-form if $\omega$ is. Also, if you have a metric, there exists a natural volume element ($n$-form in an $n$-dimensional manifold) given by $\sqrt{g} \, dx^1 \wedge \ldots \wedge dx^n$. You can easily check that you actually recover all regular integral properties and integral theorems you recognise from vector calculus using this formalism and this volume element.

 

[phoenix95]

Yup, that worked. Thanks

 

[stevendaryl]

In general, a $p$-form is the natural description of a directed $p$-dimensional hypervolume element...

@john baez, in his old set of articles about mathematical physics, said that integration actually required a pseudo-p form

https://groups.google.com/forum/#!original/sci.physics.research/aiMUJrOjE8A/jGy2N3IaajwJ

"Good," growled the Wiz. "So, listen up: You can
only integrate an n-form over a smooth n-dimensional manifold
if it is equipped with an ORIENTATION. You may be so used
to this that you've come to accept the orientation as an inevitable
prerequisite for integration. But it's not true! Integration
of pseudo n-forms works perfectly fine on any smooth manifold,
even an unoriented or unorientable one. It's only if you make
the mistake of trying to integrate an N-FORM" - he practically
spat the term out in disgust - "that you'll need an orientation.
And all the orientation does is let you convert your n-form to
a pseudo n-form! Correcting one bad move with another..." He
trailed off, grimacing at the folly of the world.​

 

[phoenix95]

I hope it's OK to get back to this. Did the theory of forms evolve with the aim of a co-ordinate free definition? Or the aim was something else and in the end turned out to be so? What is was the aim in the latter case? Could you guys please point me where I can read about this more?

 

[stevendaryl]

According to Wikipedia, they started with Elie Cartan, the founder of differential geometry. https://en.wikipedia.org/wiki/Differential_form#History

It seems to me that the concept is needed if you are going to have multi-dimensional integrals with non-Cartesian coordinates, or the special case: computing the "volume" of an n-dimensional analog of a parallelogram.