Mathematics
Using Polar Coordinates to Simplify Area Integrals
Quick fact
The Jacobian factor 'r' in the polar area element r dr dθ is the same 'r' that appears in the formula for the circumference of a circle (2πr).
Why this is interesting
You can calculate the area of a circle easily, so why does integrating over that circle feel so hard in x and y? The answer lies in choosing the right map for the job.