We write and which simplifies into , but why is differential area now ?
Also on a sidenote, if I rewrite as .
(I know that .)
Polar coordinate system - Wikipedia, the free encyclopedia
You can also draw a picture of a polar rectangle and calculate is area directly this is a very informal justification
Finally you need to calculate the new limits of integration in Polar coordinates.
Sketch the first quadrant and now decided what values and need to vary between to sweep out the first quadrant.
I hope this helps
Most textbooks that cover vector calculus have a derivation of the jacobian. I suggest you visit your library.