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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.

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,

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.

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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