Jacobian determinant of a direction vector as a function of a twodimensional vector?

I have an integral on the following form:

where is a two-dimensional vector. However, I rather want the integral on the form

where and i.e. is an extension of from two dimensions to three dimensions where 1 has been added as a third element, and is the unit hemisphere in which can exist according to its definition. We can see that while the differentials in the two integrals, and , have a different number of dimensions, the domains, ans , are both two-dimensional, which makes changing from one of the integrals to the other possible.

Normally when you change differential you use the Jacobian in the following way

where

But in my case, when is two-dimensional but is three-dimensional, how do I carry out the corresponding transformation? Since the Jacobian is the size of the hypervolume that is generated by all partial derivatives of , the closest I can get to a Jacobian, since the Jacobian itself cannot be formed in this case (the two vectors need to have the same number of dimensions), is

where is an infinitesimal surface area on and is the infinitesimal surface area on that gives rise to . So my final integral should look like

Now, I know intuitively that this is the way I should do the transformation, but how can I show it mathematically in the simplest way? How can I show that this is my "Jacobian" (although it isn't really a Jacobian) and that I really should use it?