Editing Material derivative (section)

==Definition==

The material derivative is defined for any [[tensor field]] ''y'' that is ''macroscopic'', with the sense that it depends only on position and time coordinates, {{math|1=''y'' = ''y''('''x''', ''t'')}}:
<math display="block">\frac{\mathrm{D} y}{\mathrm{D}t} \equiv \frac{\partial y}{\partial t} + \mathbf{u}\cdot\nabla y,</math>
where {{math|∇''y''}} is the [[covariant derivative]] of the tensor, and {{math|'''u'''('''x''', ''t'')}} is the [[flow velocity]]. Generally the convective derivative of the field {{math|'''u'''·∇''y''}}, the one that contains the covariant derivative of the field, can be interpreted both as involving the [[Streamline (fluid dynamics)|streamline]] [[tensor derivative (continuum mechanics)|tensor derivative]] of the field {{math|'''u'''·(∇''y'')}}, or as involving the streamline [[directional derivative]] of the field {{math|('''u'''·∇) ''y''}}, leading to the same result.<ref>{{Cite book | last=Emanuel | first=G. | title=Analytical fluid dynamics | publisher=CRC Press | year=2001 | edition=second | isbn=0-8493-9114-8 |pages=6–7 }}</ref> 
Only this spatial term containing the flow velocity describes the transport of the field in the flow, while the other describes the intrinsic variation of the field, independent of the presence of any flow. Confusingly, sometimes the name "convective derivative" is used for the whole material derivative {{math|''D''/''Dt''}}, instead for only the spatial term {{math|'''u'''·∇}}.<ref name=Batchelor/> The effect of the time-independent terms in the definitions are for the scalar and tensor case respectively known as [[advection]] and convection.

===Scalar and vector fields===
For example, for a macroscopic [[scalar field]] {{math|''φ''('''x''', ''t'')}} and a macroscopic [[vector field]] {{math|'''A'''('''x''', ''t'')}} the definition becomes:
<math display="block">\begin{align}
     \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{align}</math>

In the scalar case {{math|∇''φ''}} is simply the [[gradient]] of a scalar, while {{math|∇'''A'''}} is the covariant derivative of the macroscopic vector (which can also be thought of as the [[Jacobian matrix]] of {{math|'''A'''}} as a function of {{math|'''x'''}}).  
In particular for a scalar field in a three-dimensional [[Cartesian coordinate system]] {{math|(''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>)}}, the components of the velocity {{math|'''u'''}} are {{math|''u''<sub>1</sub>, ''u''<sub>2</sub>, ''u''<sub>3</sub>}}, and the convective term is then:
<math display="block"> \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}.</math>