How to prove $dxdy = r dr d \theta$? $x = r \cos \theta$, $y = r \sin \theta$
I got $dx = \cos \theta dr - r \sin \theta d \theta $ 
$ dy = \sin \theta dr + r \cos \theta d \theta$
How to get $dx dy = r dr d \theta$??
I saw the same question Rigorous proof that $dx dy=r\ dr\ d\theta$.
But I am not getting where vectors are coming in to the picture thanks.
 A: How to get $dx\;dy=r\;dr\;dθ$?
I suggest you take a look at Advanced calculus of several variables by C.H. Edwards. In Section 5 of Chapter IV, we can read something like this:

The student has undoubtedly seen change of variables formulas such as
  $$\iint f(x,y)\;dx\;dy=\iint f(r\cos\theta,r\sin\theta)\;r\;dr\;d\theta$$
  which result from changes from rectangular coordinates to polar coordinates. The appearance of the factor $r$ in the formula is sometimes "explained" by mythical pictures, such as figure below, in which it is alleged that $dA$ is an "infinitesimal" rectangle with sides $dr$ and $r\;d\theta$, and therefore has area $r \;dr\; d\theta$. In this section we shall give a mathemtaically acceptable explanation of the origin of such factors in the transformation of multiple integrals from one coordinate system to another. 


EDIT
For details about the Edwards approach, I refer the reader to this post.
A: A piece of an annulus swept out by a change of angle $\Delta \theta$ and a change of radius $\Delta r$, starting from a point given by $(r,\theta)$, has area $\Delta \theta \int_r^{r+\Delta r} s ds = \Delta \theta \frac{(r+\Delta r)^2-r^2}{2} = \Delta \theta \left ( r \Delta r + \frac{\Delta r^2}{2} \right )$. (This is computed by integrating the length of circular arcs.)
As $\Delta r \to 0$ the second term is asymptotically much smaller than the first, which heuristically justifies the change of variables formula. Showing that this procedure, which is equivalent to the more general procedure based on the Jacobian determinant, actually makes integrals do the correct thing takes some more work. The details can be found in a typical undergraduate real analysis text.
A: It is not so complicated. At first, direct multiplication of your two right hand quantities and simplification gives the result!
EDIT1:
Differential/infinitesmal area as an approximation in the following, is reckoned as:
$$  d\theta \,(R_o+R_i)/2 \approx  dy,\, (R_o -R_i)  \approx   dx,\, A  \approx  dx \, dy $$

And secondly/basically (geometrically) what we mean by area directly is by differential lengths multiplication of a rectangle sides of length $ dx \approx dr $ and height $ dy \approx r d\theta $ which should be appreciated by definition.  
