Question

When changing an integral to polar coordinates, DxDy becomes rDrDtheta. Can anyone give a satisfying explanation for where this r comes from?

Answer 1

I can't seem to prove it mathematically.
http://mathforum.org/library/drmath/view/74707.html
the incremental area is \(dx*dy\) in Cartesian coordinate and \(r*dr*d\theta\) in polar coordinate.

Answer 2

and since it serves as variable of integration in polar coordinate system, it should do fine.

Answer 3

That does seem to clear it up to me a bit. When looking at dxdy, you can see that it gives a rectangle with measurable area and that that area depends on dx and dy. When looking at DrDtheta, there should be a similar rectangle. Dr gives a side of that rectangle. Dtheta, by istelf, does not. The effect theta has on the area of the rectangle depends on r. So the two sidelengths we get for the rectangle are Dr and r*Dtheta.

Answer 4

one is like vector: dr = r2 - r1, and the other is like an arc or circle = \(r \theta \)

Answer 5

There is a much more clearer picture
http://math.stackexchange.com/questions/37044/explain-iint-mathrm-dx-mathrm-dy-iint-r-mathrm-d-alpha-mathrm-dr
and there's a theory beyond my understanding
http://math.stackexchange.com/questions/37044/explain-iint-mathrm-dx-mathrm-dy-iint-r-mathrm-d-alpha-mathrm-dr