Editing Material derivative (section)

===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>