Consider a one to one transformation of a $3$-$D$ volume from variable $(x,y,z)$ to $(t,u,v)$:
$$\iiint_V dx\ dy\ dz=\int_{v_1}^{v_2}\int_{u_1}^{u_2}\int_{t_1}^{t_2} \dfrac{\partial(x,y,z)}{\partial(t,u,v)} dt\ du\ dv$$
$(1)$ Now for a particular three dimensional volume, is it possible that one or more of the partial derivatives of the Jacobian of transformation doesn't exist at any of the points in the domain of integration?
$(2)$ If so, shall we compute the integral using improper integrals?