Mathematics
The Role of the Jacobian in Multivariable Change of Variables
Quick fact
The Jacobian determinant is the factor that tells you how much a small area or volume expands or shrinks when you change coordinates. For a transformation from (x,y) to (u,v), the Jacobian is the determinant of the matrix of partial derivatives, and it multiplies the differential element du dv to give the true physical size.
Why this is interesting
You've probably computed integrals in polar coordinates and noticed a mysterious extra 'r' multiplied into the area element. Why does that 'r' suddenly appear, and what does it have to do with stretching space?