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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.

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 definitión of velocity field, the chain rule and inditial notation.

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 they definición of velocity field, they chain rule and inditial notation.

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.

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

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 they definición of velocity field, they chain rule and inditial notation.