# 物質導數

## 其他名稱

• 對流導數（convective derivative）[5]
• 隨體導數（derivative following the motion）[1]
• 水動力導數（hydrodynamic derivative）[1]
• 拉格朗日導數（Lagrange derivative）[6]
• 隨質點導數（particle derivative）[7]
• 隨質導數（substantial derivative）[1]
• 實質導數（substantive derivative）[8]
• 斯托克斯導數（Stokes derivative）[8]
• 全導數（total derivative）[1][9]，雖然物質導數實際上是全導數的特殊個案[9]

## 定義

${\displaystyle {\frac {\mathrm {D} y}{\mathrm {D} t}}\equiv {\frac {\partial y}{\partial t}}+\mathbf {u} \cdot \nabla y,}$

### 純量和向量場

{\displaystyle {\begin{aligned}{\frac {\mathrm {D} \varphi }{\mathrm {D} t}}&\equiv {\frac {\partial \varphi }{\partial t}}+\mathbf {u} \cdot \nabla \varphi ,\\[3pt]{\frac {\mathrm {D} \mathbf {A} }{\mathrm {D} t}}&\equiv {\frac {\partial \mathbf {A} }{\partial t}}+\mathbf {u} \cdot \nabla \mathbf {A} .\end{aligned}}}

${\displaystyle \mathbf {u} \cdot \nabla \varphi =u_{1}{\frac {\partial \varphi }{\partial x_{1}}}+u_{2}{\frac {\partial \varphi }{\partial x_{2}}}+u_{3}{\frac {\partial \varphi }{\partial x_{3}}}.}$

## 發展

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\varphi (\mathbf {x} ,t)={\frac {\partial \varphi }{\partial t}}+{\dot {\mathbf {x} }}\cdot \nabla \varphi .}$

${\displaystyle {\dot {\mathbf {x} }}\equiv {\frac {\mathrm {d} \mathbf {x} }{\mathrm {d} t}},}$

${\displaystyle {\dot {\mathbf {x} }}=\mathbf {u} .}$

${\displaystyle {\frac {\mathrm {D} \varphi }{\mathrm {D} t}}={\frac {\partial \varphi }{\partial t}}+\mathbf {u} \cdot \nabla \varphi .}$

## 正交座標系

${\displaystyle [\left(\mathbf {u} \cdot \nabla \right)\mathbf {A} ]_{j}=\sum _{i}{\frac {u_{i}}{h_{i}}}{\frac {\partial A_{j}}{\partial q^{i}}}+{\frac {A_{i}}{h_{i}h_{j}}}\left(u_{j}{\frac {\partial h_{j}}{\partial q^{i}}}-u_{i}{\frac {\partial h_{i}}{\partial q^{j}}}\right),}$

${\displaystyle h_{i}={\sqrt {g_{ii}}}.}$

${\displaystyle (\mathbf {u} \cdot \nabla )\mathbf {A} ={\begin{pmatrix}\displaystyle u_{x}{\frac {\partial A_{x}}{\partial x}}+u_{y}{\frac {\partial A_{x}}{\partial y}}+u_{z}{\frac {\partial A_{x}}{\partial z}}\\\displaystyle u_{x}{\frac {\partial A_{y}}{\partial x}}+u_{y}{\frac {\partial A_{y}}{\partial y}}+u_{z}{\frac {\partial A_{y}}{\partial z}}\\\displaystyle u_{x}{\frac {\partial A_{z}}{\partial x}}+u_{y}{\frac {\partial A_{z}}{\partial y}}+u_{z}{\frac {\partial A_{z}}{\partial z}}\end{pmatrix}}.}$

## 參考資料

1. Bird, R.B.; Stewart, W.E.; Lightfoot, E.N. Transport Phenomena Revised Second. John Wiley & Sons. 2007: 83. ISBN 978-0-470-11539-8.
2. Batchelor, G.K. An Introduction to Fluid Dynamics. Cambridge University Press. 1967: 72–73. ISBN 0-521-66396-2.
3. ^ Trenberth, K. E. Climate System Modeling. Cambridge University Press. 1993: 99. ISBN 0-521-43231-6.
4. ^ Majda, A. Introduction to PDEs and Waves for the Atmosphere and Ocean. Courant Lecture Notes in Mathematics 9. American Mathematical Society. 2003: 1. ISBN 0-8218-2954-8.
5. ^ Ockendon, H.; Ockendon, J.R. Waves and Compressible Flow. Springer. 2004: 6. ISBN 0-387-40399-X.
6. ^ Mellor, G.L. Introduction to Physical Oceanography. Springer. 1996: 19. ISBN 1-56396-210-1.
7. ^ Stoker, J.J. Water Waves: The Mathematical Theory with Applications. Wiley. 1992: 5. ISBN 0-471-57034-6.
8. Granger, R.A. Fluid Mechanics. Courier Dover Publications. 1995: 30. ISBN 0-486-68356-7.
9. Landau, L.D.; Lifshitz, E.M. Fluid Mechanics. Course of Theoretical Physics 6 2nd. Butterworth-Heinemann. 1987: 3–4 & 227. ISBN 0-7506-2767-0.
10. ^ Emanuel, G. Analytical fluid dynamics second. CRC Press. 2001: 6–7. ISBN 0-8493-9114-8.
11. ^ Eric W. Weisstein. Convective Operator. MathWorld. [2008-07-22]. （原始内容存档于2016-03-03）.