a question

if we change from cartesian to polar coordinates in a double integral, dx dy becomes r dr dtheta. where does the r come from?

to change from one coordinate system to another, you not only have to substitute the variables of one in terms of the other, you also have to take the determinant of the jacobian matrix, which relates one coordinate system to the other.

In the case of x and y -> r and theta, where x=rCos(theta) and y=rSin(theta) this is

| dx       dy     |   | d  rCos(theta)    d rSin(theta)    |
| dr       dr     |   | dr                dr               |
|                 | = |                                    |
| dx       dy     |   | d     rCos(theta) d     rSin(theta)|
| dtheta   dtheta |   | dtheta            dtheta           |

= Cos(theta) * rCos(theta) - ( Sin(theta * (-) rSin(theta) )

= r(Cos^2(theta) + Sin^2(theta)) = r