Jacobian may be expressed on terms of material coordinates $X_p$ by using Levi-Chivita tensor.
$$J=e_{ijk}\frac{\partial x_1}{\partial X_i}\frac{\partial x_2}{\partial X_j}\frac{\partial x_3}{\partial X_k}.$$
$$\frac{DJ}{Dt}=e_{ijk}\left(\frac{\partial v_1}{\partial x_1}\frac{\partial x_1}{\partial  X_i}\frac{\partial x_2}{\partial X_j}\frac{\partial x_3}{\partial X_k}+\frac{\partial x_1}{\partial X_i}\frac{\partial v_2}{\partial x_2}\frac{\partial x_2}{\partial X_j}\frac{\partial x_3}{\partial X_k}+\frac{\partial x_1}{\partial X_i}\frac{\partial x_2}{\partial X_j}\frac{\partial v_3}{\partial x_3}\frac{\partial x_3}{\partial X_k}\right)$$
$$\frac{DJ}{Dt}=
\left(e_{ijk}\frac{\partial x_1}{\partial X_i}\frac{\partial x_2}{\partial X_j}\frac{\partial x_3}{\partial X_k}\right)\frac{\partial v_p}{\partial x_p}.$$
$$\frac{DJ}{Dt}=J\nabla\cdot\textbf{v}.$$
In obtaining this equation, I have used the definition of velocity field, they chain rule and inditial notation.