My question concerns a detail in the transformation of integrals. In one dimension, given the $$\int_a^bF(x)\mathrm d x,$$ we may perform the substitution $x=f(t)$ under appropriate conditions, which changes the parameters of the integral, so that it now appears as $$\int_A^BG(t)\color{red}{f'(t)}\mathrm d t.$$
However, when we go to higher dimensional domains, and we wish to perform analogous transformations, the new form of the integral involves, in place of $f'(t)$ above, an expression involving the partials of the substitution. For example, in two dimensions, if we wish to perform a change of the parameters of the $$\iint_R F(x,y)\mathrm d A,$$ we may perform $x=f(u,v),y=g(u,v)$ to get $$\iint_S G(u,v)\color{green}{(x_uy_v-x_vy_u)}\mathrm d A.$$ That's what we should get, but we instead use the absolute value of the expression in green (if it doesn't vanish.) My question is this: Why is it that we must ensure that this expression (usually put in determinant form and called a jacobian) is positive, especially as we usually just use its one-dimensional analogue without bothering about its sign?
Thank you.