Dimbulb, I don't think you need to split your integrals up the way you have.

It's sufficient to note that:

and

both represent area integrals over the same region (manifold), namely the

plane.

Also, you seem to be trying to change

x into

r and

y into

, which isn't kosher - you can't (generally) change coordinates pairwise like this.

If I recall my calculus correctly (and that's definitely an

if!), to change coordinates in a regular area (or volume, etc.) integral, you need to (a) ensure that the integration is occurring over the same exact region, (b) change the variables in the term being integrated, and (c) insert

det(J) into the integral, where

J is the Jacobian matrix of the coordinate transformation.

i.e.

where

Intuitively

can be understood as the ratio of the infinitesimal areas/volumes d

x : d

x'. For example, in the case of changing from cartesian coordinates

dx dy into polar coordinates

,

dr has the same scale as

dx or

dy, but

scales proportionally to

r, so at any given point

.

It appears that this is what the Grassman product notation provides "for free", in that the rules for coordinate transforms of 1-forms automatically involve calculation of the elements of

J, and then calculating out the wedge-product automatically yeilds its determinant, so when changing coordinates in an integral involving

p-forms, you don't need to worry about step (c).

Like Dimbulb, though, I'll confess I'm a little unclear about the step involving the combining two regular integrals to get a "wedge" integral, and then the later step of integrating out

from the wedge product to get a regular integral again... it seems OK, and obviously works in this case, but it's not clear if there are caveats in more general cases...