Editing Conservation law (section)

==Differential forms==
{{See also|conservation form|continuity equation}}
In [[continuum mechanics]], the most general form of an exact conservation law is given by a [[continuity equation]]. For example, conservation of electric charge {{math|''q''}} is
<math display="block">\frac{\partial \rho}{\partial t} = - \nabla \cdot \mathbf{j} \,</math>
where {{math|∇⋅}} is the [[divergence]] operator, {{math|''ρ''}} is the density of {{math|''q''}} (amount per unit volume), {{math|'''j'''}} is the flux of {{math|''q''}} (amount crossing a unit area in unit time), and {{mvar|t}} is time.

If we assume that the motion '''u''' of the charge is a continuous function of position and time, then
<math display="block">\begin{align}
 \mathbf{j} &= \rho \mathbf{u} \\
 \frac{\partial \rho}{\partial t} &= - \nabla \cdot (\rho \mathbf{u}) \,.
\end{align}</math>

In one space dimension this can be put into the form of a homogeneous first-order [[nonlinear partial differential equation|quasilinear]] [[hyperbolic equation]]:<ref name="Toro">{{cite book | first=E.F. | last=Toro | title=Riemann Solvers and Numerical Methods for Fluid Dynamics | publisher=Springer-Verlag | year=1999 | isbn=978-3-540-65966-2| chapter = Chapter 2. Notions on Hyperbolic PDEs}}</ref>{{rp|p=43}}
<math display="block"> y_t + A(y) y_x = 0 </math>
where the dependent variable {{math|''y''}} is called the ''density'' of a ''conserved quantity'', and {{math|''A''(''y'')}} is called the ''[[current Jacobian]]'', and the [[Notation for differentiation#Partial derivatives|subscript notation for partial derivatives]] has been employed. The more general inhomogeneous case:
<math display="block"> y_t + A(y) y_x = s </math>
is not a conservation equation but the general kind of [[balance equation]] describing a [[dissipative system]]. The dependent variable {{math|''y''}} is called a ''nonconserved quantity'', and the inhomogeneous term {{math|''s''(''y'',''x'',''t'')}} is the-''[[Divergence|source]]'', or [[dissipation]]. For example, balance equations of this kind are the momentum and energy [[Navier-Stokes equations]], or the [[entropy#Entropy balance equation for open systems|entropy balance]] for a general [[isolated system]].

In the '''one-dimensional space''' a conservation equation is a first-order [[nonlinear partial differential equation|quasilinear]] [[hyperbolic equation]] that can be put into the ''advection'' form:
<math display="block"> y_t + a(y) y_x = 0 </math>
where the dependent variable {{math|''y''(''x'',''t'')}} is called the density of the ''conserved'' (scalar) quantity, and {{math|''a''(''y'')}} is called the '''current coefficient''', usually corresponding to the [[partial derivative]] in the conserved quantity of a [[current density]] of the conserved quantity {{math|''j''(''y'')}}:<ref name="Toro" />{{rp|p=43}}
<math display="block"> a(y) = j_y (y)</math>

In this case since the [[chain rule]] applies:
<math display="block"> j_x = j_y (y) y_x = a(y) y_x </math>
the conservation equation can be put into the current density form:
<math display="block"> y_t + j_x (y) = 0 </math>

In a '''space with more than one dimension''' the former definition can be extended to an equation that can be put into the form:
<math display="block"> y_t + \mathbf a(y) \cdot \nabla y = 0 </math>

where the ''conserved quantity'' is {{math|''y''('''r''',''t'')}}, {{math|⋅}} denotes the [[scalar product]], {{math|∇}} is the [[nabla symbol|nabla]] operator, here indicating a [[gradient]], and {{math|''a''(''y'')}} is a vector of current coefficients, analogously corresponding to the [[divergence]] of a vector current density associated to the conserved quantity {{math|'''j'''(''y'')}}:
<math display="block"> y_t + \nabla \cdot \mathbf j(y) = 0 </math>

This is the case for the [[continuity equation]]:
<math display="block"> \rho_t + \nabla \cdot (\rho \mathbf u) = 0 </math>

Here the conserved quantity is the [[mass]], with [[density]] {{math|''ρ''('''r''',''t'')}} and current density {{math|''ρ'''''u'''}}, identical to the [[momentum conservation|momentum density]], while {{math|'''u'''('''r''', ''t'')}} is the [[flow velocity]].

In the '''general case''' a conservation equation can be also a system of this kind of equations (a [[vector equation]]) in the form:<ref name="Toro" />{{rp|p=43}}
<math display="block"> \mathbf y_t + \mathbf A(\mathbf y) \cdot \nabla \mathbf y = \mathbf 0 </math>
where {{math|'''y'''}} is called the ''conserved'' ('''vector''') quantity, {{math|∇''y''}} is its [[gradient]], {{math|'''0'''}} is the [[zero vector]], and {{math|'''A'''('''y''')}} is called the [[Jacobian matrix and determinant|Jacobian]] of the current density. In fact as in the former scalar case, also in the vector case '''A'''('''y''') usually corresponding to the Jacobian of a [[current density matrix]] {{math|'''J'''('''y''')}}:
<math display="block"> \mathbf A( \mathbf y) = \mathbf J_{\mathbf y} (\mathbf y)</math>
and the conservation equation can be put into the form:
<math display="block"> \mathbf y_t + \nabla \cdot \mathbf J (\mathbf y)= \mathbf 0 </math>

For example, this the case for Euler equations (fluid dynamics). In the simple incompressible case they are:
<math display="block">
\nabla\cdot \mathbf u = 0 \, , \qquad
\frac{\partial \mathbf u}{\partial t} + \mathbf u \cdot \nabla \mathbf u + \nabla s = \mathbf{0},
</math>

where:
*{{math|'''''u'''''}} is the [[flow velocity]] [[Vector (geometric)|vector]], with components in a N-dimensional space {{math|''u''<sub>1</sub>, ''u''<sub>2</sub>, ..., ''u<sub>N</sub>''}},
*{{math|''s''}} is the specific [[pressure]] (pressure per unit [[density]]) giving the [[Linear differential equation|source term]],

{{See also|Euler equations (fluid dynamics)}}

It can be shown that the conserved (vector) quantity and the current density matrix for these equations are respectively:

<math display="block">
{\mathbf y} = \begin{pmatrix} 1 \\  \mathbf u \end{pmatrix}; \qquad
{\mathbf J} = \begin{pmatrix}\mathbf u\\ \mathbf u \otimes \mathbf u + s \mathbf I\end{pmatrix};\qquad
</math>

where <math>\otimes</math> denotes the [[outer product]].