| There is a philosophical reason to distinguish forms, manifolds, and integration. When you talk about integrating a form on a manifold you concentrate on the result: the number you obtain. You almost do not notice that the form and the manifold exist somewhat independently. If you think of expressing forms via coordinates, you already need to have a manifold (so also a coordinate system) in place to even "define" the form. However this is not necessarily the case. Let me be more specific: Suppose your ambient space is $\R^3$ and you are looking at a vector field (let us say your space is full of water and the vector field models the velocity of the movement of water at every point). The vector field $V$ is a $1$-form, it exists. Now suppose you insert a membrane (2d surface) into the water and want to compute how much water flows through it at any given moment in time. This is the "flow" of $V$ through your surface $S$. If you go and look how to do this there are intuitive pictures and the computation reduces to
1) parameterize the 2d surface using 2 variables $(u,v)$
2) compute some partial derivatives of the parameterization
3) wedge product them
4) take the dot product with $V$
5) integrate in $u$ and $v$.
At first this seems like magic but whoever is explaining the procedure draws a bunch of pictures to explain why this is reasonable and tries to convince you. Usually they eventually manage. However this is only part of the story. You see, you have a map that inputs $V$(the vector field) and $S$ the surface and spits out a number. Furthermore this map is intuitively "continuous" in the sense that if you change $V$ a bit or $S$ a bit you do not expect the result to change too much. However if you try to prove this or explain this at any mathematical level, you run into trouble!!! The reason is that the way you defined integrating the vector field DEPENDS on the parameterization, and worse, it depends on it at the first step of your procedure. If you have to membranes that are "close" how can you even think that their parameterizations be "close". You can't! Even the SAME surface can have drastically different parameterizations. So clearly you need to abstract away the coordinates so you can talk about continuity, stability, perturbation. Let us get back to abstract definitions. You know that you can integrate 2-forms on 2-manifolds (2d surfaces). You are used to having a 2-form DEFINED on a 2-manifold (so you don't really see the difference between integration and 2-forms). However we do know that we have this rather standard procedure of computing the flow of a vector field (1-form) through a 2-manifold (2d surface). How so? It seems that for whatever reason a vector field is ALSO a 2-form. And it is a 2-form just floating around R^3 in the same way a vector field (the velocity of water) exists independently of whether you are computing how much of it is flowing through a given surface. So how is this the case? This is exactly an instance of Hodge duality. Since the ambient space $\R^3$ has a volume form (3-form) there is an intrinsic association from $k$ forms to $3-k$ forms (specifically, given a $k$ form the associated $3-k$ form is that unique $3-k$ form such that wedged with the original gives you the volume form). So there you go! Given a vector field you have an associated 2-form in $\R^3$ that is there, by itself, without needing any 2-manifold to justify its existence. In practice if $V=(V_x,V_y,V_z)$ then the two form is $V_x dy dz + V_y dz dx + V_z dx dy$. And if by chance it encounters a 2d surface it can naturally be integrated through it. The Hodge duality above actually expresses in a very concise form the multiple points on HOW to compute the flow (the procedure we started with). |