令 為區域 的參考組態,令其運動及形變梯度為

令 .
則目前組態及參考組態的積分有以下的關係
![\int _{{\Omega (t)}}{\mathbf {f}}({\mathbf {x}},t)~{\text{dV}}=\int _{{\Omega _{0}}}{\mathbf {f}}[{\boldsymbol {\varphi }}({\mathbf {X}},t),t]~J({\mathbf {X}},t)~{\text{dV}}_{0}=\int _{{\Omega _{0}}}{\hat {{\mathbf {f}}}}({\mathbf {X}},t)~J({\mathbf {X}},t)~{\text{dV}}_{0}~.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a49b5532c9a31af8181b530f93ce95d8880ab0d)
That this derivation is for a material element is implicit in the time constancy of the reference configuration: it is constant in material coordinates. 針對體積積分的微分定義為
\cfrac{\mathrm{d}}{\mathrm{d}t}\left( \int_{\Omega(t)} \mathbf{f}(\mathbf{x},t)~\text{dV}\right) =
\lim_{\Delta t \rightarrow 0} \cfrac{1}{\Delta t}
\left(\int_{\Omega(t + \Delta t)} \mathbf{f}(\mathbf{x},t+\Delta t)~\text{dV} -
\int_{\Omega(t)} \mathbf{f}(\mathbf{x},t)~\text{dV}\right) ~.
將上式轉換為對參考組態的積分,可得

因為 和時間無關,可得
![{\begin{aligned}{\cfrac {{\mathrm {d}}}{{\mathrm {d}}t}}\left(\int _{{\Omega (t)}}{\mathbf {f}}({\mathbf {x}},t)~{\text{dV}}\right)&=\int _{{\Omega _{0}}}\left[\lim _{{\Delta t\rightarrow 0}}{\cfrac {{\hat {{\mathbf {f}}}}({\mathbf {X}},t+\Delta t)~J({\mathbf {X}},t+\Delta t)-{\hat {{\mathbf {f}}}}({\mathbf {X}},t)~J({\mathbf {X}},t)}{\Delta t}}\right]~{\text{dV}}_{0}\\&=\int _{{\Omega _{0}}}{\frac {\partial }{\partial t}}[{\hat {{\mathbf {f}}}}({\mathbf {X}},t)~J({\mathbf {X}},t)]~{\text{dV}}_{0}\\&=\int _{{\Omega _{0}}}\left({\frac {\partial }{\partial t}}[{\hat {{\mathbf {f}}}}({\mathbf {X}},t)]~J({\mathbf {X}},t)+{\hat {{\mathbf {f}}}}({\mathbf {X}},t)~{\frac {\partial }{\partial t}}[J({\mathbf {X}},t)]\right)~{\text{dV}}_{0}\end{aligned}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dccf7b716b8f34767f136efbceb2832faa96de7e)
現在, 的時間導數為

因此
![{\begin{aligned}{\cfrac {{\mathrm {d}}}{{\mathrm {d}}t}}\left(\int _{{\Omega (t)}}{\mathbf {f}}({\mathbf {x}},t)~{\text{dV}}\right)&=\int _{{\Omega _{0}}}\left({\frac {\partial }{\partial t}}[{\hat {{\mathbf {f}}}}({\mathbf {X}},t)]~J({\mathbf {X}},t)+{\hat {{\mathbf {f}}}}({\mathbf {X}},t)~J({\mathbf {X}},t)~{\boldsymbol {\nabla }}\cdot {\mathbf {v}}({\mathbf {x}},t)\right)~{\text{dV}}_{0}\\&=\int _{{\Omega _{0}}}\left({\frac {\partial }{\partial t}}[{\hat {{\mathbf {f}}}}({\mathbf {X}},t)]+{\hat {{\mathbf {f}}}}({\mathbf {X}},t)~{\boldsymbol {\nabla }}\cdot {\mathbf {v}}({\mathbf {x}},t)\right)~J({\mathbf {X}},t)~{\text{dV}}_{0}\\&=\int _{{\Omega (t)}}\left({\dot {{\mathbf {f}}}}({\mathbf {x}},t)+{\mathbf {f}}({\mathbf {x}},t)~{\boldsymbol {\nabla }}\cdot {\mathbf {v}}({\mathbf {x}},t)\right)~{\text{dV}}\end{aligned}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8fb723181cce1321f237c980778b89bc72717f7d)
其中 為 的 材料導數 ,現在材料導數為
![{\dot {{\mathbf {f}}}}({\mathbf {x}},t)={\frac {\partial {\mathbf {f}}({\mathbf {x}},t)}{\partial t}}+[{\boldsymbol {\nabla }}{\mathbf {f}}({\mathbf {x}},t)]\cdot {\mathbf {v}}({\mathbf {x}},t)~.](https://wikimedia.org/api/rest_v1/media/math/render/svg/0527b00dc98359f37547adb6ed77d26e9b65d5de)
因此
![{\cfrac {{\mathrm {d}}}{{\mathrm {d}}t}}\left(\int _{{\Omega (t)}}{\mathbf {f}}({\mathbf {x}},t)~{\text{dV}}\right)=\int _{{\Omega (t)}}\left({\frac {\partial {\mathbf {f}}({\mathbf {x}},t)}{\partial t}}+[{\boldsymbol {\nabla }}{\mathbf {f}}({\mathbf {x}},t)]\cdot {\mathbf {v}}({\mathbf {x}},t)+{\mathbf {f}}({\mathbf {x}},t)~{\boldsymbol {\nabla }}\cdot {\mathbf {v}}({\mathbf {x}},t)\right)~{\text{dV}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ccadc7aec02eeff04b0876b7d6c97d41f277add)
或者

利用以下的恆等式

可得

利用高斯散度定理及恆等式
,可得

|