Editing Control volume (section)

== Substantive derivative ==
{{Main|Material derivative}}
Computations in continuum mechanics often require that the regular time [[Derivative|derivation]] operator 
<math>d/dt\;</math> 
is replaced by the [[substantive derivative]] operator
<math>D/Dt</math>.
This can be seen as follows.

Consider a bug that is moving through a volume where there is some [[scalar field|scalar]], 
e.g. [[pressure]], that varies with time and position: 
<math>p=p(t,x,y,z)\;</math>.

If the bug during the time interval from 
<math>t\;</math> 
to 
<math>t+dt\;</math> 
moves from 
<math>(x,y,z)\;</math> 
to 
<math>(x+dx, y+dy, z+dz),\;</math>
then the bug experiences a change <math>dp\;</math> in the scalar value,
:<math>dp = \frac{\partial p}{\partial t}dt 
+ \frac{\partial p}{\partial x}dx 
+ \frac{\partial p}{\partial y}dy 
+ \frac{\partial p}{\partial z}dz</math>
(the [[total derivative|total differential]]). If the bug is moving with a [[velocity]]
<math>\mathbf v = (v_x, v_y, v_z),</math> 
the change in particle position is 
<math>\mathbf v dt = (v_xdt, v_ydt, v_zdt),</math> 
and we may write
:<math>\begin{alignat}{2}
dp & 
= \frac{\partial p}{\partial t}dt 
+ \frac{\partial p}{\partial x}v_xdt 
+ \frac{\partial p}{\partial y}v_ydt 
+ \frac{\partial p}{\partial z}v_zdt \\ &
= \left(
\frac{\partial p}{\partial t}
+ \frac{\partial p}{\partial x}v_x 
+ \frac{\partial p}{\partial y}v_y 
+ \frac{\partial p}{\partial z}v_z
\right)dt \\ &
= \left(
\frac{\partial p}{\partial t} 
+ \mathbf v \cdot\nabla p
\right)dt. \\
\end{alignat}</math>

where <math>\nabla p</math> is the [[gradient]] of the scalar field ''p''. So:

:<math>\frac{d}{dt} = \frac{\partial}{\partial t} + \mathbf v \cdot\nabla.</math>

If the bug is just moving with the flow, the same formula applies, but now the velocity vector,''v'', is [[flow velocity|that of the flow]], ''u''.
The last parenthesized expression is the substantive derivative of the scalar pressure.
Since the pressure p in this computation is an arbitrary scalar field, we may abstract it and write the substantive derivative operator as
:<math>\frac{D}{Dt} = \frac{\partial}{\partial t} + \mathbf u \cdot\nabla.</math>