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I don't know if this answer will be of much help. The problem is that, in Analysis, the standard argument for the deduction of the expressions of area and volume elements under a change of coordinates, is to argue heuristically using the dx's; these arguments give the right results, as shown by the previous answers, but the elements themselves are not rigorously defined.

The real reason for the correct results of these arguments lies not in Analysis, but in Differential Geometry instead; there the objects are differential manifolds, a deep generalization of surfaces that are locally undistinguishable from R^n, for some n; given this, we may define functions on the manifolds and generalize all the concepts of the ordinary Differential and Integral Calculus and more. This is done by defining vector spaces that are "tangent" to the manifold's points (they are, in fact, called tangent spaces).

Now, if you have a vector space V you may define its algebraic dual V* and, from these, a whole bunch of them via tensor products.

This pertains to your question because the dx's, rigorously defined, are elements of V*; in fact, they are the elements of the dual base of a "natural" base of the Tangent Space at a point, and they are called 1-forms.

Then, when you write dxdy, what you are saying is something more complicated:

dxdy = dx∧dy (it's a 2-form)

Where the "∧" is an algebraic operation called the exterior product; again, it's a generalization of the more familiar exterior product on R^3.

When you change coordinates, you change the local maps that define the manifold coordinates in your neighborhood, and it's possible to prove that the corresponding vectors in the Tangent Space and its dual at that point also change according to the differential of the coordinate transformation and, when this is particularized to R^n, this differential is precisely the Jacobian Matrix; then the properties of the exterior product and the manner in which the coordinates of the linear functionals (the dx's) change when the basis is changed conspire to give you the general expression that you see in analysis.

This is very frustrating, because it's impossible to detail here, but it's my best attempt to give a glimpse of what's going on underneath these expressions. To give you one more example of the depth and power of these Differential Geometry concepts, take a look at this:

∫[M]dω = ∫[∂M]ω

Where M is a n-dimensional manifold, ∂M it's its boundary (a (n - 1)-dimensional manifold), ω is a (n - 1)-form, and dω it's its exterior differential, a n-form. This equality is a consequence of a very deep result, called de Rham's Theorem, and I write here because it's the general form of FIVE different Analysis theorems:

(1) The Fundamental Theorem of Calculus.

(2) The Integration by Parts Formula.

(3) Green's Theorem in R^2.

(4) Gauss's Theorem in R^3.

(5) Stokes Theorem on R^3.

They're all there. Sorry about the rambling.

The "little box" can be viewed as having a radial dimension and a tangential dimension. The radial part has measure dr, and the tangential part has measure r dθ, so it has approximate area dr * r dθ=r*dr*dθ.

If you formally calculate the determinant of the Jacobian J of transformation given by x = r cos θ and y = r sin θ, the result is |J|=r, so the elementary area given by |J|drdθ is then r*dr*dθ.

yes, we have to use the Jacobian determinant.

dx dy =

│∂x/∂r ∂x/∂θ│

│∂y/∂r ∂y/∂θ│

* dr dθ

= rdrdθ