I have a really hard time understanding when to use the Jacobian (in the change of variables formula) when I'm calculating integrals. Every time I think I get it I always stumble upon a case when I shouldn't use it. It seems rather arbitrary, for example:
I'm looking at the definition/derivation of the surface integral of the vector field $F$ on the area $T$: $$\int\int_T(F\cdot n)dS$$ $T$ parametrized with $r(s,t)$ with the domain $D$. With $$n = \frac{r'_s \times r'_t}{|r'_s \times r'_t|},$$ $$dS = |r'_s \times r'_t|dsdt$$ We get: $$\int\int_T(F\cdot n)dS = \int\int_DF(r(s,t))\cdot \frac{r'_s \times r'_t}{|r'_s \times r'_t|}\cdot|r'_s \times r'_t|dsdt$$ Which nets $$\int\int_DF(r(s,t))\cdot (r'_s \times r'_t)dsdt$$
My question is: Why aren't we using the jacobian according to the change of variable formula to compensate when going from the domain $T$ to $D$?