I've just begun investigating differential forms. I have no experience in this field and no formal, university level training in mathematics, so please bear with me.(adsbygoogle = window.adsbygoogle || []).push({});

I understand that a differential form may be thought of as a family of linear functionals; more precisely, it is a function that assigns to each point of a manifold a linear functional (i.e., a member of the cotangent space at that point) mapping a tuple of k tangent vectors to a real number.

This is all good for me. My confusion arises with regard to what sorts of quantities can actually be computed using forms.

For example, I understand that, if we take our manifold to be the xy-plane in ordinary Euclidean space, a two-form, say A dx dy where A is constant, can be evaluated on a geometrical figure in this plane to give its oriented area up to the factor A (and by "evaluated" I really mean integrated over the figure). But what if I wanted to compute the surface area of a surface in general xyz-space, say the upper-hemisphere of the unit sphere? This is straightforward in vector calculus using surface integrals, but I can't see how this could be accomplished using differential forms.

To make my difficulty plain, consider an analogous but simpler problem. To find the arc length of a curve in the xy-plane, we would need to integrate the quantity [tex]\sqrt{dx^2 + dy^2}[/tex] over the curve in the language of differential forms. Clearly the product here cannot be the Grassmann (wedge) product, since those terms would then vanish. But even if a suitable definition were provided for the products, the result doesn't seem like it'd be a differential form.

My problem is that it seems to me that differential forms naturally compute areas, volumes, etc,within a tangent space, but for computing such quantities over a portion of a manifold forms are not sufficient--the use of some more general object becomes necessary.

I'd be very grateful for anyone who can help to correct any misunderstandings I might have. I'm learning this subject on my own, just for the love of it. This isn't for a class or anything.

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# Differential Forms and Areas

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