Editing Continuum mechanics (section)

===Eulerian description===
Continuity allows for the inverse of <math>\chi(\cdot)</math> to trace backwards where the particle currently located at <math>\mathbf x</math> was located in the initial or referenced configuration <math>\kappa_0(\mathcal B)</math>. In this case the description of motion is made in terms of the spatial coordinates, in which case is called the spatial description or Eulerian description, i.e. '''the current configuration is taken as the reference configuration'''.

The Eulerian description, introduced by [[d'Alembert]], focuses on the current configuration <math>\kappa_t(\mathcal B)</math>, giving attention to what is occurring at a fixed point in space as time progresses, instead of giving attention to individual particles as they move through space and time. This approach is conveniently applied in the study of [[fluid mechanics|fluid flow]] where the kinematic property of greatest interest is the rate at which change is taking place rather than the shape of the body of fluid at a reference time.{{sfn|Spencer|1980|p=83}}

Mathematically, the motion of a continuum using the Eulerian description is expressed by the mapping function

:<math>\mathbf X=\chi^{-1}(\mathbf x, t)</math>

which provides a tracing of the particle which now occupies the position <math>\mathbf x</math> in the current configuration <math>\kappa_t(\mathcal B)</math> to its original position <math>\mathbf X</math> in the initial configuration <math>\kappa_0(\mathcal B)</math>.

A necessary and sufficient condition for this inverse function to exist is that the determinant of the [[Jacobian matrix and determinant|Jacobian matrix]], often referred to simply as the Jacobian, should be different from zero. Thus,

:<math>J = \left| \frac{\partial \chi_i}{\partial X_J} \right| = \left| \frac{\partial x_i}{\partial X_J} \right| \neq 0</math>

In the Eulerian description, the physical properties <math>P_{ij\ldots}</math> are expressed as

:<math>P_{ij \ldots}=P_{ij\ldots}(\mathbf X,t)=P_{ij\ldots}[\chi^{-1}(\mathbf x,t),t]=p_{ij\ldots}(\mathbf x,t)</math>

where the functional form of <math>P_{ij \ldots}</math> in the Lagrangian description is not the same as the form of <math>p_{ij \ldots}</math> in the Eulerian description.

The material derivative of <math>p_{ij\ldots}(\mathbf x,t)</math>, using the chain rule, is then

:<math>\frac{d}{dt}[p_{ij\ldots}(\mathbf x,t)]=\frac{\partial}{\partial t}[p_{ij\ldots}(\mathbf x,t)]+ \frac{\partial}{\partial x_k}[p_{ij\ldots}(\mathbf x,t)]\frac{dx_k}{dt}</math>

The first term on the right-hand side of this equation gives the ''local rate of change'' of the property <math>p_{ij\ldots}(\mathbf x,t)</math> occurring at position <math>\mathbf x</math>. The second term of the right-hand side is the ''convective rate of change'' and expresses the contribution of the particle changing position in space (motion).

Continuity in the Eulerian description is expressed by the spatial and temporal continuity and continuous differentiability of the flow velocity field. All physical quantities are defined this way at each instant of time, in the current configuration, as a function of the vector position <math>\mathbf x</math>.