Editing Material derivative

{{Short description|Time rate of change of some physical quantity of a material element in a velocity field}}
In [[continuum mechanics]], the '''material derivative'''<ref name="BSLr2"/><ref name=Batchelor>{{cite book | first=G. K. | last=Batchelor | author-link=George Batchelor | title=An Introduction to Fluid Dynamics | year=1967 | publisher=Cambridge University Press | isbn=0-521-66396-2 | pages=72–73}}</ref> describes the time [[derivative|rate of change]] of some physical quantity (like [[heat]] or [[momentum]]) of a [[material element]] that is subjected to a space-and-time-dependent [[flow velocity|macroscopic velocity field]]. The material derivative can serve as a link between [[Continuum mechanics#Eulerian description|Eulerian]] and [[Continuum mechanics#Lagrangian description|Lagrangian]] descriptions of continuum [[Deformation (mechanics)|deformation]].<ref name=Trenberth>{{cite book | first=K. E. | last=Trenberth | author-link = Kevin Trenberth | title=Climate System Modeling | year=1993 | publisher=Cambridge University Press | isbn=0-521-43231-6 | page=99 }}</ref>

For example, in [[fluid dynamics]], the velocity field is the [[flow velocity]], and the quantity of interest might be the [[temperature]] of the fluid. In this case, the material derivative then describes the temperature change of a certain [[fluid parcel]] with time, as it flows along its [[Streamlines, streaklines, and pathlines|pathline]] (trajectory).
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==Other names==
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There are many other names for the material derivative, including:
*'''advective derivative'''<ref>{{cite book |title=Introduction to PDEs and Waves for the Atmosphere and Ocean | last=Majda |first=A. |author-link=Andrew Majda |isbn=0-8218-2954-8 |series=Courant Lecture Notes in Mathematics |volume=9 |year=2003 |publisher=American Mathematical Society |page=1 }}</ref>
*'''convective derivative'''<ref name=Ockendon>{{cite book| first1=H. |last1=Ockendon |author1-link= Hilary Ockendon |last2=Ockendon |first2=J.R.|author2-link= John Ockendon | title=Waves and Compressible Flow | publisher=Springer | year=2004 | isbn=0-387-40399-X | page=6 }}</ref>
*'''derivative following the motion'''<ref name="BSLr2"/>
*'''hydrodynamic derivative'''<ref name="BSLr2"/>
*'''Lagrangian derivative'''<ref name=Mellor>{{cite book | first=G.L. | last=Mellor | title=Introduction to Physical Oceanography | publisher=Springer | year=1996 | isbn=1-56396-210-1 |page=19 }}</ref>
*'''particle derivative'''<ref>{{cite book |title=Water Waves: The Mathematical Theory with Applications |last=Stoker |first=J.J. |author-link=James J. Stoker |isbn=0-471-57034-6 |page=5 |year=1992 |publisher=Wiley }}</ref>
*'''substantial derivative'''<ref name="BSLr2">{{cite book|last1=Bird |first1=R.B. |last2=Stewart |first2=W.E. | last3=Lightfoot |first3=E.N. |author3-link=Edwin N. Lightfoot |title=[[Transport Phenomena (book)|Transport Phenomena]] |edition=Revised Second |publisher=John Wiley & Sons |year=2007 |isbn=978-0-470-11539-8 |page=83}}</ref>
*'''substantive derivative'''<ref name=Granger>{{cite book| first=R.A. |last=Granger| title=Fluid Mechanics | publisher=Courier Dover Publications | year=1995 | isbn=0-486-68356-7 | page=30}}</ref>
*'''Stokes derivative'''<ref name=Granger/>
*'''total derivative''',<ref name="BSLr2"/><ref name="LandauLifshitz">{{Cite book | publisher = Butterworth-Heinemann | isbn = 0-7506-2767-0
| first1 = L.D. | last1 = Landau | author1-link = Lev Landau | first2 = E.M. | last2 = Lifshitz | author2-link = Evgeny Lifshitz | title = Fluid Mechanics | edition = 2nd | series = Course of Theoretical Physics | volume = 6 | year = 1987 | pages = 3–4 & 227 }}</ref> although the material derivative is actually a special case of the [[total derivative]]<ref name="LandauLifshitz"/>

==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>

==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]].

==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>

==See also==
{{Div col|colwidth=25em}}
* [[Navier–Stokes equations]]
* [[Euler equations (fluid dynamics)]]
* [[Derivative (generalizations)]]
* [[Lagrangian and Eulerian specification of the flow field]]
* [[Lie derivative]]
* [[Levi-Civita connection]]
* [[Spatial acceleration]]
* [[Spatial gradient]]
{{Div col end}}

==References==
{{Reflist}}

==Further reading==
* {{cite book
|first1=Ira M.|last1=Cohen|first2=Pijush K|last2=Kundu
|title=Fluid Mechanics|isbn=978-0-12-373735-9|publisher=[[Academic Press]]|edition=4th |year=2008 }}
* {{cite book|first1=Michael|last1=Lai|first2=Erhard|last2=Krempl|first3=David|last3=Ruben
|title=Introduction to Continuum Mechanics|isbn=978-0-7506-8560-3|publisher=Elsevier|edition=4th |year=2010 }}

[[Category:Fluid dynamics]]
[[Category:Multivariable calculus]]
[[Category:Rates]]
[[Category:Generalizations of the derivative]]