Editing Material derivative (section)

==Development==
Consider a scalar quantity {{math|1=''φ'' = ''φ''('''x''', ''t'')}}, where {{mvar|t}} is time and {{math|'''x'''}} is position. Here {{math|''φ''}} may be some physical variable such as temperature or chemical concentration. The physical quantity, whose scalar quantity is {{math|''φ''}}, exists in a continuum, and whose macroscopic velocity is represented by the vector field {{math|'''u'''('''x''', ''t'')}}.

The (total) derivative with respect to time of {{math|''φ''}} is expanded using the multivariate [[chain rule]]:
<math display="block">\frac{\mathrm{d}}{\mathrm{d} t}\varphi(\mathbf x(t), t) = \frac{\partial \varphi}{\partial t} + \dot \mathbf x \cdot \nabla \varphi.</math>

It is apparent that this derivative is dependent on the vector
<math display="block">\dot \mathbf x \equiv \frac{\mathrm{d} \mathbf x}{\mathrm{d} t},</math>
which describes a ''chosen'' path {{math|'''x'''(''t'')}} in space. For example, if <math> \dot \mathbf x= \mathbf 0</math> is chosen, the time derivative becomes equal to the partial time derivative, which agrees with the definition of a [[partial derivative]]: a derivative taken with respect to some variable (time in this case) holding other variables constant (space in this case). This makes sense because if <math>\dot \mathbf x = 0</math>, then the derivative is taken at some ''constant'' position. This static position derivative is called the Eulerian derivative.

An example of this case is a swimmer standing still and sensing temperature change in a lake early in the morning: the water gradually becomes warmer due to heating from the sun. In which case the term <math> {\partial \varphi}/{\partial t}</math> is sufficient to describe the rate of change of temperature.

If the sun is not warming the water (i.e. <math> {\partial \varphi}/{\partial t} = 0</math>), but the path {{math|'''x'''(''t'')}} is not a standstill, the time derivative of {{math|''φ''}} may change due to the path. For example, imagine the swimmer is in a motionless pool of water, indoors and unaffected by the sun. One end happens to be at a constant high temperature and the other end at a constant low temperature. By swimming from one end to the other the swimmer senses a change of temperature with respect to time, even though the temperature at any given (static) point is a constant. This is because the derivative is taken at the swimmer's changing location and the second term on the right <math> \dot \mathbf x \cdot \nabla \varphi </math> is sufficient to describe the rate of change of temperature. A temperature sensor attached to the swimmer would show temperature varying with time, simply due to the temperature variation from one end of the pool  to the other.

The material derivative finally is obtained when the path {{math|'''x'''(''t'')}} is chosen to have a velocity equal to the fluid velocity
<math display="block">\dot \mathbf x = \mathbf u.</math>

That is, the path follows the fluid current described by the fluid's velocity field {{math|'''u'''}}. So, the material derivative of the scalar {{math|''φ''}} is
<math display="block">\frac{\mathrm{D} \varphi}{\mathrm{D} t} = \frac{\partial \varphi}{\partial t} + \mathbf u \cdot \nabla \varphi.</math>

An example of this case is a lightweight, neutrally buoyant particle swept along a flowing river and experiencing  temperature changes as it does so. The temperature of the water locally may be increasing due to one portion of the river being sunny and the other in a shadow, or the water as a whole may be heating as the day progresses. The changes due to the particle's motion (itself caused by fluid motion) is called ''[[advection]]'' (or convection if a vector is being transported).

The definition above relied on the physical nature of a fluid current; however, no laws of physics were invoked (for example, it was assumed that a lightweight particle in a river will follow the velocity of the water), but it turns out that many physical concepts can be described concisely using the material derivative. The general case of advection, however, relies on conservation of mass of the fluid stream; the situation becomes slightly different if advection happens in a non-conservative medium.

Only a path was considered for the scalar above. For a vector, the gradient becomes a [[tensor derivative]]; for [[tensor]] fields we may want to take into account not only translation of the coordinate system due to the fluid movement but also its rotation and stretching. This is achieved by the [[upper convected time derivative]].