Why determinant gives area enlargement?

Hi!

Iīm trying to fully understand why the determinant gives the area enlargement when switching between variables.

This might belong in calculus, but the book says linear algebra "shows this" so here it is.

Like when we do polar coordinates: $\displaystyle x = r\cdot cos(\theta) \mbox{ and } y = r\cdot sin(\theta) $

Then we multiply with the jacobian-determinant, and get an additional $\displaystyle r $ .

Are there any good resources where I can read some more about this?

Thx