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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?

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  • $\begingroup$ Question regarding this topic, despite being very interesting, are also very technical. I answered a question of such kind here, however dealing with a non differentiable Jacobian which is not the same thing as a vanishing Jacobian. Perhaps you'll find something useful in the references. $\endgroup$ Jul 12 '19 at 18:01
  • $\begingroup$ What do you mean by vanishing Jacobian? $\endgroup$
    – Joe
    Jul 13 '19 at 15:53
  • $\begingroup$ You know that in order for the map $(x,y,z)\mapsto (t,u,v)$ to be injective, its Jacobian $J$ should be $>0$. However the change of variable formula holds also if $J=0$, say for example on a null measure set in $V$: this is a vanishing Jacobian, which however implies that the associated map is differentiable. $\endgroup$ Jul 13 '19 at 16:04

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