Change of variables theorem and the Jacobian determinant

Consider a small rectangle $R_{ij} = [u_i, u_i + \Delta u] \times [v_j, v_j + \Delta v]$ in $U$. By first-order Taylor expansion around $(u_i, v_j)$, the edges $( \Delta u, 0 )$ and $( 0, \Delta v )$ map approximately to the tangent vectors $\mathbf{a} = \Phi_u(u_i,v_j)\,\Delta u = (x_u, y_u)\,\Delta u$ and $\mathbf{b} = \Phi_v(u_i,v_j)\,\Delta v = (x_v, y_v)\,\Delta v$. The area of the parallelogram spanned by $\mathbf{a}$ and $\mathbf{b}$ in $\mathbb{R}^2$ equals $|x_u y_v - x_v y_u|\,\Delta u\,\Delta v = |\det D\Phi(u_i, v_j)|\,\Delta u\,\Delta v$ up to higher-order error $o(\Delta u\,\Delta v)$.

Substituting $\Delta A_{ij} \approx |\det D\Phi(u_i, v_j)|\,\Delta u\,\Delta v$ into the Riemann sum $\sum_{i,j} f(\Phi(u_i, v_j))\,\Delta A_{ij}$ yields $\sum_{i,j} f(\Phi(u_i, v_j))\,|\det D\Phi(u_i, v_j)|\,\Delta u\,\Delta v$. In particular, for polar coordinates $x = r\cos\theta$, $y = r\sin\theta$, we have $\det D\Phi = (\cos\theta)(r\cos\theta) - (-r\sin\theta)(\sin\theta) = r(\cos^2\theta + \sin^2\theta) = r$, giving $dx\,dy = r\,dr\,d\theta$.