Think I may have figured out the source of my confusion. Please tell me if this makes sense.
Since the Jacobian is taken for some scalarvalued or vectorvalued function with respect to another vector, we're actually considering the FUNCTION which takes one coordinate system to another.
So when I want to change an integral from Cartesian coordinates to cylindrical coordinates I have to use the following functions:
x = g( r, theta, z) = rcos(theta)
y = h(r, theta, z) = rsin(theta)
z = m(r, theta, z) = z
The functions g, h, m when aggregated into a vector are a map FROM (r, theta, z) TO (x,y,z). So even though I use said map to change my integral from Cartesian to cylindrical, the map itself is from cylindrical coordinates into (x,y,z) coordinates.
I think that's why everything seemed "reversed" to me. Have I got it now, though?
