1. 物质导数与空间时间导数及二者的联系

考虑运动变形过程中代表性物质点的物理量 $\mathbf{\Phi }$(张量) 随时间的变化率。

• 在物质描述中，$\mathbf{\Phi }$ 以 $(\vec{X},t)$ 为自变量；
• 在空间描述中，$\mathbf{\Phi }$ 以 $(\vec{x},t)$ 为自变量。

物理量 $\mathbf{\Phi }$ 随某一固定的物质点一起运动的时间变化率（称作：物质导数）可写作：
$$\frac{D\mathbf{\Phi }}{Dt}={\left(\frac{\partial \mathbf{\Phi }(\vec{X},t)}{\partial t}\right)|}_{\vec{X}}\triangleq \stackrel{\bullet }{\mathbf{\Phi }}$$
物理量 $\mathbf{\Phi }$ 在某一固定的空间坐标上的时间变化率（称作：空间时间导数/局部导数）可写作：
$${\left(\frac{\partial \mathbf{\Phi }(\vec{x},t)}{\partial t}\right)|}_{\vec{x}}\triangleq {\mathbf{\Phi }}^{\prime }$$

根据复合函数的求导法则可推出：
$$\begin{array}{rl}& \stackrel{\bullet }{\mathbf{\Phi }}={\left\{\frac{\partial \mathbf{\Phi }[\vec{x}(\vec{X},t),t]}{\partial t}\right\}|}_{\vec{X}}\\ & \\ & ={\left(\frac{\partial \mathbf{\Phi }}{\partial t}\right)|}_{\vec{x}}+\left(\frac{\partial \mathbf{\Phi }}{\partial x^r}\right){\left(\frac{\partial x^r}{\partial t}\right)|}_{\vec{X}}\\ & \\ & ={\mathbf{\Phi }}^{\prime }+\left(\frac{\partial \mathbf{\Phi }}{\partial x^r}\otimes \vec{g}{}^r\right)\cdot {\left({\vec{g}}_s\frac{\partial x^s}{\partial t}\right)|}_{\vec{X}}\\ & \\ & ={\mathbf{\Phi }}^{\prime }+\left(\mathbf{\Phi }▽\right)\cdot {\left(\frac{\partial \vec{x}}{\partial t}\right)|}_{\vec{X}}\\ & \\ & ={\mathbf{\Phi }}^{\prime }+\left(\mathbf{\Phi }▽\right)\cdot \left(\frac{\partial \vec{u}}{\partial t}\right)\\ & \\ & ={\mathbf{\Phi }}^{\prime }+\left(\mathbf{\Phi }▽\right)\cdot \vec{v}\\ & \\ & ={\mathbf{\Phi }}^{\prime }+\vec{v}\cdot \left(▽\mathbf{\Phi }\right)\end{array}$$

2. 空间坐标系相关量的物质导数

2.1. 空间坐标系基矢的物质导数

随时间变化，某一固定物质点将映射至空间坐标系中的不同位置。因此，“空间坐标系基矢的物质导数”是指：某一物质点所在处的基矢变化率。故
$$\stackrel{\bullet }{{\vec{g}}_i}=({\vec{g}}_i{)}^{\prime }+\vec{v}\cdot ▽{\vec{g}}_i=\vec{v}\cdot ▽{\vec{g}}_i=v^j\frac{\partial {\vec{g}}_i}{\partial x^j}=v^j{\Gamma }_{ij}^k{\vec{g}}_k=v^j{\Gamma }_{ij,k}{\vec{g}}^k$$
式中，${\Gamma }_{ij}^k、{\Gamma }_{ij,k}$ 分别为空间坐标系的第二类、第一类 Christoffel 符号。又由于
$$\frac{D}{Dt}({\vec{g}}_i\cdot {\vec{g}}^j)=\stackrel{\bullet }{{\vec{g}}_i}\cdot {\vec{g}}^j+{\vec{g}}_i\cdot \stackrel{\bullet }{{\vec{g}}^j}=0⟹{\vec{g}}_i\cdot \stackrel{\bullet }{{\vec{g}}^j}=-\stackrel{\bullet }{{\vec{g}}_i}\cdot {\vec{g}}^j$$
令 $\stackrel{\bullet }{{\vec{g}}^j}={\beta }_i^j{\vec{g}}^i$ ，那么：
$${\vec{g}}_i\cdot {\beta }_k^j{\vec{g}}^k={\beta }_i^j=-\stackrel{\bullet }{{\vec{g}}_i}\cdot {\vec{g}}^j=-v^k{\Gamma }_{ik}^j$$
故，
$$\stackrel{\bullet }{{\vec{g}}^j}=-v^k{\Gamma }_{ik}^j{\vec{g}}^i$$

2.2. 空间坐标系协变基矢混合积的 $\sqrt{g}$ 的物质导数

由空间坐标系基矢的物质导数可知：
$$\stackrel{\bullet }{g_{ij}}=\stackrel{\bullet }{{\vec{g}}_i}\cdot {\vec{g}}_j+{\vec{g}}_i\cdot \stackrel{\bullet }{{\vec{g}}_j}=v^r({\Gamma }_{ir}^k{g}_{kj}+{\Gamma }_{jr}^k{g}_{ki})=v^r({\Gamma }_{ir,j}+{\Gamma }_{jr,i})$$
由于，
$$\frac{1}{det([A])}[A^{\ast }]=[A{]}^{-1}$$
其中，$[A^{\ast }]$ 为 $[A]$ 的伴随矩阵。则
$$\frac{1}{g}\frac{\partial g}{\partial g_{ji}}=g^{ij}，g=det(g_{ij})$$
故，$det(g_{ij})$ 的物质导数为：
$$\stackrel{\bullet }{g}=\frac{\partial g}{\partial g_{ji}}\stackrel{\bullet }{g_{ji}}=g{g}^{ij}\stackrel{\bullet }{g_{ji}}=g{v}^r({\Gamma }_{ir}^i+{\Gamma }_{jr}^j)=2g{v}^r{\Gamma }_{ir}^i$$
式中，${\Gamma }_{ir}^i$ 为空间坐标系的第二类Christoffel 符号。进一步：
$$\stackrel{\bullet }{\sqrt{g}}=\sqrt{g}{v}^r{\Gamma }_{ir}^i$$
上式也可利用第二类Christoffel符号与协变基矢的混合积 $\sqrt{g}$ 的关系与物质导数和局部导数的关系得到：
$$\stackrel{\bullet }{\sqrt{g}}=\vec{v}\cdot (▽\sqrt{g})=v^i\frac{\partial \sqrt{g}}{\partial x^i}=\sqrt{g}{v}^r{\Gamma }_{ir}^i$$

3. 随体坐标系 {XA,t}\{X^A,t\}{XA,t} 相关量的物质导数

3.1. 随体坐标系 {XA,t}\{X^A,t\}{XA,t} 基矢的物质导数

随时间的变化，特定的物质点在随体坐标系 {XA,t}\{X^A,t\}{XA,t} 中的基矢不断改变。其协变基矢的变化率可写作：
$$\begin{array}{rl}& \stackrel{\bullet }{{\vec{C}}_A}={\left[\frac{\partial }{\partial t}\left(\frac{\partial \vec{x}}{\partial X^A}\right)\right]|}_{\vec{X}}={\left[\frac{\partial }{\partial X^A}\left(\frac{\partial \vec{x}}{\partial t}\right)\right]|}_{\vec{X}}\\ & \\ & =\frac{\partial }{\partial X^A}\left(\frac{\partial \vec{u}}{\partial t}\right)=\frac{\partial \vec{v}}{\partial X^A}=v^B|{|}_A{\vec{C}}_B\\ & \\ & =\frac{\partial \vec{v}}{\partial x^i}\frac{\partial x^i}{\partial X^A}=x_{,A}^i\frac{\partial \vec{v}}{\partial x^i}=x_{,A}^i{v}^j{|}_i{\vec{g}}_j\end{array}$$
同理可知
$$\stackrel{\bullet \bullet }{{\vec{C}}_A}=\frac{\partial \vec{a}}{\partial X^A}=a^B|{|}_A{\vec{C}}_B$$
又
$$\frac{D{\vec{C}}_A\cdot {\vec{C}}^B}{Dt}=\stackrel{\bullet }{{\vec{C}}_A}\cdot {\vec{C}}^B+{\vec{C}}_A\cdot \stackrel{\bullet }{{\vec{C}}^B}=0$$
故
$$\stackrel{\bullet }{{\vec{C}}^A}=-(\stackrel{\bullet }{{\vec{C}}_B}\cdot {\vec{C}}^A){\vec{C}}^B=-v^A|{|}_B{\vec{C}}^B=-X_{,j}^A{v}^j{|}_i{\vec{g}}^i$$

3.2. 随体坐标系 {XA,t}\{X^A,t\}{XA,t} 协变基矢混合积的 $\sqrt{C}$ 的物质导数

$${\stackrel{\bullet }{C}}_{AB}={\stackrel{\bullet }{\vec{C}}}_A\cdot {\vec{C}}_B+{\vec{C}}_A\cdot {\stackrel{\bullet }{\vec{C}}}_B=v_B|{|}_A+v_A|{|}_B$$
又
$$\frac{1}{C}\frac{\partial C}{\partial C_{BA}}=\stackrel{-1}{C}{}^{AB}，C=det(C_{AB})$$
则
$$\stackrel{\bullet }{C}=\frac{\partial C}{\partial C_{BA}}{\stackrel{\bullet }{C}}_{AB}=C\stackrel{-1}{C}{}^{AB}{\stackrel{\bullet }{C}}_{AB}=C(v^A|{|}_A+v^B|{|}_B)=2C{v}^A|{|}_A$$
进一步知：
$$\stackrel{\bullet }{\sqrt{C}}=\frac{1}{2\sqrt{C}}\stackrel{\bullet }{C}=\sqrt{C}{v}^A|{|}_A$$

3.3. $\mathcal{J}$ 的物质导数

由于，
$$\mathcal{J}=det(\mathbf{F})=\sqrt{\frac{C}{G}}$$
故，
$$\stackrel{\bullet }{\mathcal{J}}=\frac{\stackrel{\bullet }{\sqrt{C}}}{\sqrt{G}}=\frac{\sqrt{C}}{\sqrt{G}}{v}^A|{|}_A=\mathcal{J}{v}^A|{|}_A=\mathcal{J}▽\cdot \vec{v}$$

4. 任意张量在空间坐标系与随体坐标系 {XA,t}\{X^A,t\}{XA,t} 中的物质导数

以三阶张量为例：
$$\mathbf{\Phi }={\Phi }_{\bullet \bullet k}^{ij}{\vec{g}}_i\otimes {\vec{g}}_j\otimes {\vec{g}}^k={\Phi }_{\bullet \bullet M}^{AB}{\vec{C}}_A\otimes {\vec{C}}_B\otimes {\vec{C}}^M$$
则
$$\begin{array}{rl}& \stackrel{\bullet }{\mathbf{\Phi }}=\stackrel{\bullet }{\Phi }{}_{\bullet \bullet M}^{AB}{\vec{C}}_A\otimes {\vec{C}}_B\otimes {\vec{C}}^M+{\Phi }_{\bullet \bullet M}^{AB}\stackrel{\bullet }{{\vec{C}}_A}\otimes {\vec{C}}_B\otimes {\vec{C}}^M+{\Phi }_{\bullet \bullet M}^{AB}{\vec{C}}_A\otimes \stackrel{\bullet }{{\vec{C}}_B}\otimes {\vec{C}}^M+{\Phi }_{\bullet \bullet M}^{AB}{\vec{C}}_A\otimes {\vec{C}}_B\otimes \stackrel{\bullet }{\vec{C}}{}^M\\ & \\ & =(\stackrel{\bullet }{\Phi }{}_{\bullet \bullet M}^{AB}+{\Phi }_{\bullet \bullet M}^{NB}{v}^A|{|}_N+{\Phi }_{\bullet \bullet M}^{AN}{v}^B|{|}_N-{\Phi }_{\bullet \bullet N}^{AB}{v}^N|{|}_M){\vec{C}}_A\otimes {\vec{C}}_B\otimes {\vec{C}}^M\end{array}$$
或
$$\begin{array}{rl}& \stackrel{\bullet }{\mathbf{\Phi }}=\stackrel{\bullet }{\Phi }{}_{\bullet \bullet k}^{ij}{\vec{g}}_i\otimes {\vec{g}}_j\otimes {\vec{g}}^k+{\Phi }_{\bullet \bullet k}^{ij}{\stackrel{\bullet }{\vec{g}}}_i\otimes {\vec{g}}_j\otimes {\vec{g}}^k+{\Phi }_{\bullet \bullet k}^{ij}{\vec{g}}_i\otimes {\stackrel{\bullet }{\vec{g}}}_j\otimes {\vec{g}}^k+{\Phi }_{\bullet \bullet k}^{ij}{\vec{g}}_i\otimes {\vec{g}}_j\otimes \stackrel{\bullet }{\vec{g}}{}^k\\ & \\ & =(\stackrel{\bullet }{\Phi }{}_{\bullet \bullet k}^{ij}+\Phi {}_{\bullet \bullet k}^{sj}{v}^r{\Gamma }_{rs}^i+\Phi {}_{\bullet \bullet k}^{is}{v}^r{\Gamma }_{rs}^j-\Phi {}_{\bullet \bullet s}^{ij}{v}^r{\Gamma }_{rk}^s){\vec{g}}_i\otimes {\vec{g}}_j\otimes {\vec{g}}^k\\ & \\ & =[(\Phi {}_{\bullet \bullet k}^{ij}{)}^{\prime }+(v^r\Phi {}_{\bullet \bullet k,r}^{ij}+\Phi {}_{\bullet \bullet k}^{sj}{v}^r{\Gamma }_{rs}^i+\Phi {}_{\bullet \bullet k}^{is}{v}^r{\Gamma }_{rs}^j-\Phi {}_{\bullet \bullet s}^{ij}{v}^r{\Gamma }_{rk}^s)]{\vec{g}}_i\otimes {\vec{g}}_j\otimes {\vec{g}}^k\\ & \\ & =[(\Phi {}_{\bullet \bullet k}^{ij}{)}^{\prime }+v^r\Phi {}_{\bullet \bullet k}^{ij}{|}_r)]{\vec{g}}_i\otimes {\vec{g}}_j\otimes {\vec{g}}^k\\ & \\ & ={\mathbf{\Phi }}^{\prime }+\vec{v}\cdot ▽\mathbf{\Phi }\end{array}$$