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The distortion factor between size in $uv$space and size in $xy$ space is called the Jacobian. The following video explains what the Jacobian is, how it accounts for distortion, and how it appears in the changeofvariable formula.
Let's see why the Jacobian is the distortion factor in general for a mapping $${\bf \Phi} : (u,\, v) \ \to \ (x(u,\,v),\, y(u,\,v)) \ = \ x(u,\,v)\, {\bf i} +y(u,\,v)\, {\bf j}\,, $$ making good use of all the vector calculus we've developed so far. Let $Q = [a,\,a+h]\times [c,\,c+k]$ be a rectangle in the $uv$plane and ${\bf \Phi}(Q)$ its image in the $xy$plane as shown in Then $${\bf u} \ = \ {\bf \Phi}(a+h,\,c)  {\bf \Phi}(a,\,c)\,, \qquad {\bf v} \ = \ {\bf \Phi}(a,\,c+k)  {\bf \Phi}(a,\,c)\,.$$ The area of the parallelogram spanned by ${\bf u} = u_1 {\bf i} + u_2 {\bf j}$ and ${\bf v} = v_1 {\bf i} + v_2 {\bf j}$ is the determinant $\left  \begin{matrix} u_1 & v_1 \cr u_2 & v_2 \end{matrix}\right $. By the definition of partial derivatives, $$\frac{{\bf \Phi}(a+h,\,c)  {\bf \Phi}(a,\,c)}{h} \ \approx \ \frac{\partial {\bf \Phi}}{\partial u}\Big_{(a,c)}\ = \ \frac{\partial x}{\partial u}\Big_{(a,c)}\, {\bf i} + \frac{\partial y}{\partial u}\Big_{(a,c)}\, {\bf j} \,,$$ $$\frac{{\bf \Phi}(a,\,c+k)  {\bf \Phi}(a,\,c)}{k} \ \approx \ \frac{\partial {\bf \Phi}}{\partial v}\Big_{(a,c)}\ = \ \frac{\partial x}{\partial v}\Big_{(a,c)}\, {\bf i} + \frac{\partial y}{\partial v}\Big_{(a,c)}\, {\bf j}\,.$$ We then compute $$\hbox{area}(\Phi(Q)) \approx \left  \begin{matrix} u_1 & v_1 \cr u_2 & v_2 \end{matrix}\right  \ \approx \ \left  \begin{matrix} h \frac{\partial x}{\partial u} & k \frac{\partial x}{\partial v} \cr h \frac{\partial y}{\partial u} & k \frac{\partial y}{\partial v} \end{matrix} \right  \ = \ hk \left  \begin{matrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \cr \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v}\end{matrix} \right . $$
So why didn't we see an absolute value in the changeofvariables formula in one dimension? This had to do with the way we write the limits of integration.
