Editing Material derivative (section)

==Orthogonal coordinates==
It may be shown that, in [[orthogonal coordinates]], the {{math|''j''}}-th component of the convection term of the material derivative of a [[vector field]] <math>\mathbf{A}</math> is given by<ref>{{cite web
| url = http://mathworld.wolfram.com/ConvectiveOperator.html
| title = Convective Operator
| author = Eric W. Weisstein
| author-link = Eric W. Weisstein
| publisher = [[MathWorld]]
| access-date = 2008-07-22
}}</ref>
<math display="block">[\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),
</math>

where the {{math|''h''<sub>''i''</sub>}} are related to the [[metric tensor]]s by <math>h_i = \sqrt{g_{ii}}.</math>

In the special case of a three-dimensional [[Cartesian coordinate system]] (''x'', ''y'', ''z''), and {{math|'''A'''}} being a 1-tensor (a vector with three components), this is just:
<math display="block">(\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} =
\frac{\partial (A_x, A_y, A_z)}{\partial (x, y, z)}\mathbf{u}
</math>

where <math>\frac{\partial(A_x, A_y, A_z)}{\partial(x, y, z)}</math> is a [[Jacobian matrix]].

There is also a [[vector_calculus_identities#Vector-dot-Del_Operator|vector-dot-del identity]] and the material derivative for a vector field <math>\mathbf A</math> can be expressed as:

:<math> {\displaystyle (\mathbf {A} \cdot \nabla )\mathbf {A} = {\frac {1}{2}}\nabla |\mathbf {A} |^{2}-\mathbf {A} \times (\nabla \times \mathbf {A} )={\frac {1}{2}}\nabla |\mathbf {A} |^{2}+(\nabla \times \mathbf {A} )\times \mathbf {A} }.</math>