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| {{continuum mechanics}}
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| A '''continuity equation''' in physics is an [[equation]] that describes the transport of a [[conserved quantity]]. Since [[mass]], [[energy]], [[momentum]], [[electric charge]] and other natural quantities are conserved under their respective appropriate conditions, a variety of physical phenomena may be described using continuity equations.
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| Continuity equations are a stronger, local form of [[conservation law]]s. For example, it is true that "[[conservation of energy|the total energy in the universe is conserved]]". But this statement does not immediately rule out the possibility that energy could disappear from Earth while simultaneously appearing in another galaxy. A stronger statement is that energy is ''locally'' conserved: Energy can neither be created nor destroyed, ''nor'' can it "teleport" from one place to another—it can only move by a continuous flow. A continuity equation is the mathematical way to express this kind of statement.
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| Continuity equations more generally can include "source" and "sink" terms, which allow them to describe quantities that are often but not always conserved, such as the density of a molecular species which can be created or destroyed by chemical reactions. In an everyday example, there is a continuity equation for the number of living humans; it has a "source term" to account for people giving birth, and a "sink term" to account for people dying.
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| Any continuity equation can be expressed in an "integral form" (in terms of a [[Flux#Flux_as_a_surface_integral|flux integral]]), which applies to any finite region, or in a "differential form" (in terms of the [[divergence]] operator) which applies at a point.
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| Continuity equations underlie more specific [[transport equation]]s such as the [[convection–diffusion equation]], [[Boltzmann transport equation]], and [[Navier-Stokes equations]].
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| ==General equation==
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| ===Preliminary description===
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| [[File:Continuity eqn open surface.svg|390pxright|thumb|Illustration of how flux '''j''' passes through open curved surfaces ''S'' (d'''S''' is differential [[vector area]]).]]
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| [[File:Continuity eqn closed surface.svg|300px|right|thumb|Illustration of how flux '''j''' passes through closed surfaces ''S''<sub>1</sub> and ''S''<sub>2</sub>. The surface area elements shown are d'''S'''<sub>1</sub> and d'''S'''<sub>2</sub>, and the flux is integrated over the whole surface. Yellow dots are sources, red dots are sinks, the blue lines are the flux lines of ''q''.]]
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| As stated above, the idea behind the continuity equation is the flow of some property, such as [[mass]], [[energy]], [[electric charge]], [[momentum]], and even [[probability]], through [[surface]]s from one region of space to another. The surfaces, in general, may either be open or closed, real or imaginary, and have an arbitrary shape, but are fixed for the calculation (i.e. not time-varying, which is appropriate since this complicates the maths for no advantage). Let this property be represented by just one [[Scalar (mathematics)|scalar]] variable, ''q'', and let the [[volume]] density of this property (the amount of ''q'' per unit volume ''V'') be ''ρ'', and the [[Union (set theory)|union]] of all surfaces be denoted by ''S''. Mathematically, ''ρ'' is a ratio of two infinitesimal quantities:
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| :<math>\rho = \frac{d q}{d V},</math>
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| which has the dimension [quantity][L]<sup>−3</sup> (where L is length).
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| There are different ways to conceive the continuity equation:
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| #either the flow of [[particle]]s carrying the quantity ''q'', described by a [[velocity field]] '''v''', which is also equivalent to a [[flux]] '''j''' of ''q'' (a [[Vector-valued function|vector function]] describing the flow per unit area per unit time of ''q''), or
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| #in the cases where a velocity field is not useful or applicable, the flux '''j''' of the quantity ''q'' only (no association with velocity).
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| In each of these cases, the transfer of ''q'' occurs as it passes through two surfaces, the first ''S''<sub>1</sub> and the second ''S''<sub>2</sub>.
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| [[File:Continuity eqn quantities.svg|250px|"250px"|right|thumb|Illustration of ''q'', ''ρ'', and '''j''', and the effective flux due to carriers of ''q''. ''ρ'' is the amount of ''q'' per unit volume (in the box), '''j''' represents the flux (blue flux lines) and ''q'' is carried by the particles (yellow).]]
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| The flux '''j''' should represent some flow or transport, which has dimensions [quantity][T]<sup>−1</sup>[L]<sup>−2</sup>. In cases where particles/carriers of quantity ''q'' are moving with velocity '''v''', such as particles of mass in a fluid or charge carriers in a conductor, '''j''' can be related to '''v''' by:
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| :<math>\mathbf{j} = \rho \mathbf{v} .</math>
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| This relation is only true in situations where there are particles moving and carrying ''q'' - it can't always be applied. To illustrate this: if '''j''' is electric [[current density]] ([[electric current]] per unit [[Area (mathematics)|area]]) and ''ρ'' is the [[charge density]] (charge per unit volume), then the velocity of the [[charge carriers]] is '''v'''. However - if '''j''' is [[heat flux|heat flux density]] (heat energy per unit time per unit area), then even if we let ''ρ'' be the heat [[energy density]] (heat energy per unit volume) it does ''not'' imply the "velocity of heat" is '''v''' (this makes no sense, and is not practically applicable). In the latter case only '''j''' (with ''ρ'') may be used in the continuity equation.
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| ===Elementary vector form===
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| Consider the case when the surfaces are flat and [[Plane (geometry)|planar]] [[Cross section (geometry)|cross-section]]s. For the case where a velocity field can be applied, [[dimensional analysis]] leads to this form of the '''continuity equation''':
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| :<math> \rho_1 \mathbf{v}_1 \cdot \mathbf{S}_1 = \rho_2 \mathbf{v}_2 \cdot \mathbf{S}_2 </math> | |
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| where
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| *the left hand side is the initial amount of ''q'' flowing per unit time through surface ''S''<sub>1</sub>, the right hand side is the final amount through surface ''S''<sub>2</sub>,
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| *'''S'''<sub>1</sub> and '''S'''<sub>2</sub> are the vector areas for the surfaces ''S''<sub>1</sub> and ''S''<sub>2</sub> respectively.
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| Notice the [[dot product]]s <math> \mathbf{v}_1 \cdot \mathbf{S}_1, \, \mathbf{v}_2 \cdot \mathbf{S}_2 \,\!</math> are [[volumetric flow rate]]s of ''q''. The dimension of each side of the equation is [quantity][L]<sup>−3</sup>•[L][T]<sup>−1</sup>•[L]<sup>2</sup> = [quantity][T]<sup>−1</sup>. For the more general cases, independent of whether a velocity field can be used or not, the continuity equation becomes:
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| :<math> \mathbf{j}_1 \cdot \mathbf{S}_1 = \mathbf{j}_2 \cdot \mathbf{S}_2 </math>
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| This has exactly the same dimensions as the previous version. The relation between '''j''' and '''v''' allows us to pass from the velocity version to this flux equation, but ''not'' always the other way round (as explained above - velocity fields are not always applicable). These results can be generalized further to curved surfaces by reducing the vector surfaces into infinitely many differential surface elements (that is '''S''' → d'''S'''), then integrating over the surface:
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| :<math> \int\!\!\!\!\int_{S_1} \rho_1\mathbf{v}_1 \cdot d\mathbf{S}_1 = \int\!\!\!\!\int_{S_2} \rho_2\mathbf{v}_2 \cdot d\mathbf{S}_2 </math>
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| more generally still: | |
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| {{Equation box 1
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| |indent=:
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| |equation = <math> \int\!\!\!\!\int_{S_1} \mathbf{j}_1 \cdot d\mathbf{S}_1 = \int\!\!\!\!\int_{S_2} \mathbf{j}_2 \cdot d\mathbf{S}_2 </math>
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| |cellpadding
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| |border
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| |border colour = #50C878
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| |background colour = #ECFCF4}}
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| in which
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| * <math>\int\!\!\!\!\int_S d\mathbf{S} \equiv \int\!\!\!\!\int_S \mathbf{\hat{n}}dS </math> denotes a [[surface integral]] over the surface ''S'',
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| * <math>\mathbf{\hat{n}}</math> is the outward-pointing [[unit normal]] to the surface ''S''
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| N.B: the scalar area ''S'' and [[vector area]] '''S''' are related by <math>d\mathbf{S} = \mathbf{\hat{n}}dS</math>. Either notations may be used interchangeably.
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| ===Differential form===
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| The differential form for a general continuity equation is (using the same ''q'', ''ρ'' and '''j''' as above):
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| {{Equation box 1
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| |indent=:
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| |equation=<math>\frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{j} = \sigma\,</math>
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| |cellpadding
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| |border
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| |border colour = #0073CF
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| |background colour=#F5FFFA}}
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| where
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| *∇• is [[divergence]],
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| *''t'' is [[time]],
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| *''σ'' is the generation of ''q'' per unit volume per unit time. Terms that generate (''σ'' > 0) or remove (''σ'' < 0) ''q'' are referred to as a "sources" and "sinks" respectively.
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| This general equation may be used to derive any continuity equation, ranging from as simple as the volume continuity equation to as complicated as the [[Navier–Stokes equations]]. This equation also generalizes the [[advection equation]]. Other equations in physics, such as [[Gauss's law|Gauss's law of the electric field]] and [[Gauss's law for gravity]], have a similar mathematical form to the continuity equation, but are not usually called by the term "continuity equation", because '''j''' in those cases does not represent the flow of a real physical quantity. | |
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| In the case that ''q'' is a [[conservation law|conserved quantity]] that cannot be created or destroyed (such as [[energy]]), this translates to ''σ'' = 0, and the continuity equation is:
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| :<math>\frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{j} = 0\,</math>
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| ==={{anchor|Integral form|integral form}} Integral form=== <!-- anchor to refer to here from other pages -->
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| [[File:SurfacesWithAndWithoutBoundary.svg|right|thumb|250px|In the integral form of the continuity equation, ''S'' is any imaginary [[closed surface]] that fully encloses a volume ''V'', like any of the surfaces on the left. ''S'' can ''not'' be a surface with boundaries that do not enclose a volume, like those on the right. (Surfaces are blue, boundaries are red.)]]
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| By the [[divergence theorem]] (see below), the continuity equation can be rewritten in an equivalent way, called the "integral form":
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| {{Equation box 1
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| |indent=:
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| |equation={{oiint
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| | preintegral = <math>\frac{d q}{d t} + </math>
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| | intsubscpt = <math>\scriptstyle S</math>
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| | integrand = <math>\mathbf{j} \cdot d\mathbf{S} = \Sigma</math>
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| }}
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| |cellpadding
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| |border
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| |border colour = #0073CF
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| |background colour=#F5FFFA}}
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| where
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| *''S'' is a surface as described above - except this time it has to be a [[closed surface]] that encloses a volume ''V'',
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| *{{oiint
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| | preintegral =
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| | intsubscpt = <math>\scriptstyle S</math>
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| | integrand=<math>d\mathbf{S}</math>}} denotes a [[surface integral]] over a closed surface,
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| *<math>\int\!\!\!\int\!\!\!\int_V \, dV </math> denotes a [[volume integral]] over ''V''.
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| *<math>q = \int\!\!\!\!\int\!\!\!\!\int_V \rho \, dV </math> is the total amount of ''ρ'' in the volume ''V'';
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| *<math>\Sigma = \int\!\!\!\!\int\!\!\!\!\int_V \sigma \, dV</math> is the total generation (negative in the case of removal) per unit time by the sources and sinks in the volume ''V'',
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| In a simple example, ''V'' could be a building, and ''q'' could be the number of people in the building. The surface ''S'' would consist of the walls, doors, roof, and foundation of the building. Then the continuity equation states that the number of people in the building increases when people enter the building (an inward flux through the surface), decreases when people exit the building (an outward flux through the surface), increases when someone in the building gives birth (a "source" where ''σ'' > 0), and decreases when someone in the building dies (a "sink" where ''σ'' < 0).
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| ===Derivation and equivalence===
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| The differential form can be derived from first principles as follows.
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| ====Derivation of the differential form====
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| Suppose first an amount of quantity ''q''(''t'') at time ''t'' is contained in a region of volume ''V'', bounded by a closed surface ''S'', as described above. The surface and volume of the region can be taken to be independent of time ''t''. Denoting the volume density of ''q'' by ''ρ''('''r''', ''t''), then ''q''(''t'') can be written as the [[volume integral]] of ''ρ'' in the region:
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| :<math>q(t) = \int\!\!\!\!\int\!\!\!\!\int_V \rho(\mathbf{r},t) dV.</math> | |
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| The rate of change of ''q'' is simply the time derivative of ''q'', so [[differentiating under the integral sign]] with respect to time:
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| :<math> \frac{dq(t)}{dt} = \frac{d}{dt} \int\!\!\!\!\int\!\!\!\!\int_V \rho(\mathbf{r},t) dV = \int\!\!\!\!\int\!\!\!\!\int_V \frac{\partial \rho(\mathbf{r},t) }{\partial t} dV .</math>
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| The derivative is changed from the total to partial as it enters the integral because the integrand (density ''ρ'') is not only a function of time, but also of coordinates. The rate of change of ''q'' can also be expressed as a sum of the flow through the surface ''S'' taken with the minus sign (the flow is from inside to outside) and the rate of production of ''q'':
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| :{{oiint
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| | preintegral = <math>\frac{dq(t)}{dt} =-</math>
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| | intsubscpt = <math>\scriptstyle S</math>
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| | integrand=<math>\mathbf{j}(\mathbf{r},t)\cdot d\mathbf{S} + \Sigma(t).</math>}}
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| Now equating these expressions:
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| :{{oiint
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| | preintegral = <math> - </math>
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| | intsubscpt = <math>\scriptstyle S</math>
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| | integrand=<math>\mathbf{j}(\mathbf{r},t)\cdot d\mathbf{S} + \Sigma(t)=\int\!\!\!\!\int\!\!\!\!\int_V \frac{\partial \rho(\mathbf{r},t)}{\partial t} dV,</math>}}
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| Using the [[divergence theorem]] on the left-hand side:
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| :<math>-\int\!\!\!\!\int\!\!\!\!\int_V \nabla\cdot\mathbf{j}(\mathbf{r},t) dV + \int\!\!\!\!\int\!\!\!\!\int_V \sigma(\mathbf{r},t) dV = \int\!\!\!\!\int\!\!\!\!\int_V \frac{\partial \rho(\mathbf{r},t)}{\partial t} dV. </math>
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| Since the volume ''V'' is arbitrary chosen this is only true if the integrands are equal, which directly leads to the differential continuity equation:
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| :<math>\begin{align} & \nabla\cdot\mathbf{j}(\mathbf{r},t) = - \frac{\partial \rho(\mathbf{r},t)}{\partial t} + \sigma(\mathbf{r},t), \\
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| & \nabla\cdot\mathbf{j} + \frac{\partial \rho}{\partial t} = \sigma \quad \rightleftharpoons \quad \nabla\cdot(\rho \mathbf{v}) + \frac{\partial \rho}{\partial t} = \sigma.\\
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| \end{align}</math>
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| Either form may be useful and quoted, both can appear in hydrodynamics and electromagnetism, but for quantum mechanics and energy conservation, only the first is used. Therefore the first is more general.
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| ====Equivalence between differential and integral form====
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| Starting from the differential form which is for unit volume, multiplying throughout by the infinitesimal volume element d''V'' and integrating over the region gives the total amounts quantities in the volume of the region (per unit time):
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| :<math>\begin{align} & \int\!\!\!\!\int\!\!\!\!\int_V \frac{\partial \rho(\mathbf{r},t)}{\partial t} dV + \int\!\!\!\!\int\!\!\!\!\int_V \nabla \cdot \mathbf{j}(\mathbf{r},t) dV = \int\!\!\!\!\int\!\!\!\!\int_V \sigma(\mathbf{r},t) dV \\
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| & \frac{d}{d t} \int\!\!\!\!\int\!\!\!\!\int_V \rho(\mathbf{r},t)dV + \int\!\!\!\!\int\!\!\!\!\int_V \nabla \cdot \mathbf{j}(\mathbf{r},t) dV = \Sigma(t) \end{align} \,\!</math>
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| again using the fact that ''V'' is constant in shape for the calculation, so it is independent of time and the time derivatives can be freely moved out of that integral, ordinary derivatives replace partial derivatives since the integral becomes a function of time only (the integral is evaluated over the region - so the spatial variables become removed from the final expression and ''t'' remains the only variable).
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| Using the [[divergence theorem]] on the left side
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| :{{oiint | |
| | preintegral = <math>\frac{d q(t)}{d t} + </math>
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| | intsubscpt = <math>\scriptstyle S</math>
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| | integrand=<math>\mathbf{j}(\mathbf{r},t)\cdot d\mathbf{S} = \Sigma(t)</math>}}
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| which is the integral form.
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| ====Equivalence between elementary and integral form====
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| Starting from
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| :<math> \int\!\!\!\!\int_{S_1} \mathbf{j}_1(\mathbf{r},t) \cdot d\mathbf{S}_1 = \int\!\!\!\!\int_{S_2} \mathbf{j}_2(\mathbf{r},t) \cdot d\mathbf{S}_2 </math>
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| the surfaces are equal (since there is only one closed surface), so '''S'''<sub>1</sub> = '''S'''<sub>2</sub> = '''S''' and we can write:
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| :<math> \int\!\!\!\!\int_{S} \mathbf{j}_1(\mathbf{r},t) \cdot d\mathbf{S} = \int\!\!\!\!\int_{S} \mathbf{j}_2(\mathbf{r},t) \cdot d\mathbf{S} </math>
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| The left hand side is the flow rate of quantity ''q'' occurring inside the closed surface ''S''. This must be equal to
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| :<math> \int\!\!\!\!\int_{S} \mathbf{j}_1(\mathbf{r},t) \cdot d\mathbf{S} = \Sigma(t) - \frac{dq(t)}{dt} </math>
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| since some is produced by sources, hence the positive term ''Σ'', but some is also leaking out by passing through the surface, implied by the negative term -d''q''/d''t''. Similarly the right hand side is the amount of flux passing through the surface and out of it, so
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| :{{oiint
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| | preintegral = <math> \int\!\!\!\!\int_S \mathbf{j}_2(\mathbf{r},t) \cdot d\mathbf{S} = </math>
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| | intsubscpt = <math>\scriptstyle S</math>
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| | integrand=<math>\mathbf{j}(\mathbf{r},t) \cdot d\mathbf{S}</math>}}
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| Equating these:
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| :{{oiint
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| | preintegral = <math> \Sigma(t) - \frac{dq(t)}{dt} = </math>
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| | intsubscpt = <math>\scriptstyle S</math>
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| | integrand=<math> \mathbf{j}(\mathbf{r},t) \cdot d\mathbf{S} </math>}}
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| :{{oiint
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| | preintegral = <math>\frac{dq}{dt}+</math>
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| | intsubscpt = <math>\scriptstyle S</math>
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| | integrand=<math> \mathbf{j} \cdot d\mathbf{S} =\Sigma </math>}}
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| which is the integral form again.
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| ===Four-current formulation===
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| {{see also|Four-vector}}
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| The density ''ρ'' and current '''j''' as above can be combined into a general four-dimensional current (not necessarily an electromagnetic [[4-current]]);
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| :<math>J^\mu = \left(c \rho, \mathbf{j} \right)</math>
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| where ''c'' is the [[speed of light]] and ''μ'' is an [[index notation|index]] labelling the [[spacetime]] [[dimension]].
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| The four-[[divergence]] of this current is: | |
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| :<math>\partial_\mu J^\mu = \frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{j} </math>
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| where ∂<sub>μ</sub> is the [[four-gradient]], so the continuity equation becomes:
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| :<math>\partial_\mu J^\mu = \sigma</math>
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| which is simpler to write and is [[Lorentz invariant]].
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| == Electromagnetism ==
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| {{Main|Charge conservation}}
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| In [[electromagnetic theory]], the continuity equation is an empirical law expressing (local) [[charge conservation]]. Mathematically it is an automatic consequence of [[Maxwell's equations]], although charge conservation is more fundamental than Maxwell's equations. It states that the [[divergence]] of the [[current density]] '''J''' (in [[amperes]] per square meter) is equal to the negative rate of change of the [[charge density]] ''ρ'' (in [[coulomb]]s per cubic metre),
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| :<math> \nabla \cdot \mathbf{J} = - {\partial \rho \over \partial t} </math>
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| :{| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
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| !Consistency with Maxwell's equations
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| |-
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| One of [[Maxwell's equations]], [[Ampère's law|Ampère's law (with Maxwell's correction)]], states that
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| :<math> \nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t}. </math>
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| Taking the divergence of both sides results in
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| :<math> \nabla \cdot ( \nabla \times \mathbf{H} ) = \nabla \cdot \mathbf{J} + \frac{\partial (\nabla \cdot \mathbf{D})}{\partial t}, </math>
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| but the divergence of a curl is zero, so that
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| :<math> \nabla \cdot \mathbf{J} + \frac{\partial (\nabla \cdot \mathbf{D})}{\partial t} = 0. </math>
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| Another one of Maxwell's equations, [[Gauss's law]], states that
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| :<math> \nabla \cdot \mathbf{D} = \rho,\, </math>
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| substitution into the previous equation yields the continuity equation
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| :<math> \nabla \cdot \mathbf{J} + {\partial \rho \over \partial t} = 0.\,</math>
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| |}
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| Current is the movement of charge. The continuity equation says that if charge is moving out of a differential volume (i.e. divergence of current density is positive) then the amount of charge within that volume is going to decrease, so the rate of change of charge density is negative. Therefore the continuity equation amounts to a conservation of charge.
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| If [[magnetic monopole]]s exist, there would be a continuity equation for monopole currents as well, see the monopole article for background and the duality between electric and magnetic currents.
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| == Fluid dynamics ==
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| {{see also|Mass flux|Vorticity equation}}
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| In [[fluid dynamics]], the continuity equation states that, in any [[steady state]] process, the rate at which mass enters a system is equal to the rate at which mass leaves the system.<ref name=Pedlosky>{{Cite book
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| | publisher = [[Springer Science+Business Media|Springer]]
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| | isbn = 978-0-387-96387-7
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| | last = Pedlosky
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| | first = Joseph
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| | title = Geophysical fluid dynamics
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| | year = 1987
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| | pages = 10–13
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| }}</ref><ref>Clancy, L.J.(1975), ''Aerodynamics'', Section 3.3, Pitman Publishing Limited, London</ref>
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| The differential form of the continuity equation is:<ref name=Pedlosky/>
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| :<math> {\partial \rho \over \partial t} + \nabla \cdot (\rho \mathbf{u}) = 0</math>
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| where
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| *''ρ'' is fluid [[density]],
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| *''t'' is time,
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| *'''u''' is the [[flow velocity]] [[vector field]].
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| In this context, this equation is also one of [[Euler equations (fluid dynamics)]]. The [[Navier-Stokes equations]] form a vector continuity equation describing the conservation of [[linear momentum]].
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| If ''ρ'' is a constant, as in the case of [[incompressible flow]], the mass continuity equation simplifies to a volume continuity equation:<ref name=Pedlosky/>
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| :<math>\nabla \cdot \mathbf{u} = 0,</math>
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| which means that the [[divergence]] of velocity field is zero everywhere. Physically, this is equivalent to saying that the local volume dilation rate is zero.
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| == Thermodynamics ==
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| Conservation of energy (which, in non-relativistic situations, can only be transferred, and not created or destroyed) leads to a continuity equation, an alternative mathematical statement of energy conservation to the [[Laws of thermodynamics|thermodynamic laws]].
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| Letting
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| *''u'' = local [[energy density]] (energy per unit volume),
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| *'''q''' = [[energy flux]] (transfer of energy per unit cross-sectional area per unit time) as a vector,
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| the continuity equation is:
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| :<math> \nabla \cdot \mathbf{q} + \frac{ \partial u}{\partial t} = 0</math>
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| [[Baryon number conservation]] is more fundamental than the [[first law of thermodynamics|first]] and [[second law of thermodynamics|second]] laws of thermodynamics.<ref>{{Cite book|title=Gravitation|author=J.A. Wheeler, C. Misner, K.S. Thorne|publisher=W.H. Freeman & Co|year=1973|pages=558–559|isbn=0-7167-0344-0}}</ref> If
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| *''n'' is the total [[number density]] of [[baryon]]s in the [[local coordinate system|local]] [[rest frame]] of a fluid (positive value for baryons, negative for [[antiparticle|anti]]baryons),
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| *''U''<sup>α</sup> is the [[4-velocity]] field of the baryon fluid flow,
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| then the [[4-baryon number flux]] vector is ''S''<sup>α</sup> = ''nU''<sup>α</sup>, satisfying the continuity equation:
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| :<math>\partial_\alpha S^\alpha = \partial_\alpha (nU^\alpha) = 0</math>
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| == Quantum mechanics ==<!-- This section is linked from [[Conservation law]] -->
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| In [[quantum mechanics]], the ''conservation of probability'' also yields a continuity equation. The terms in the equation require the following definitions, and are slightly less obvious than the other forms of volume densities, currents, current densities etc., so they are outlined here:
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| * The [[wavefunction]] ''Ψ'' for a single [[particle]] in [[position and momentum space|position space]] (rather than [[position and momentum space|momentum space]]), that is, a function of position '''r''' and time ''t'', ''Ψ'' = ''Ψ''('''r''', ''t'').
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| * The [[probability density function]] is:
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| :<math> \rho(\mathbf{r},t) = \Psi^{*}(\mathbf{r},t) \Psi(\mathbf{r},t) = |\Psi(\mathbf{r},t)|^2 \,\!</math>
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| * The [[probability]] of finding the particle within ''V'' at ''t'' is denoted and defined by:
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| :<math>P=P_{\mathbf{r} \in V}(t) = \int_V \Psi^{*} \Psi d V = \int_V |\Psi|^2 d V \,</math>
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| * The [[probability current]] (aka probability flux):
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| :<math> \mathbf{j}(\mathbf{r},t) = \frac{\hbar}{2mi} \left [ \Psi^{*} \left ( \nabla \Psi \right ) - \Psi \left ( \nabla \Psi^{*} \right ) \right ]. </math>
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| With these definitions the continuity equation reads:
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| :<math> \nabla \cdot \mathbf{j} + \frac{\partial \rho}{\partial t} = 0 \rightleftharpoons \nabla \cdot \mathbf{j} + \frac{\partial |\Psi|^2}{\partial t} = 0.</math>
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| Either form may be quoted. Intuitively; the above quantities indicate this represents the flow of probability. The ''chance'' of finding the particle at some position '''r''' and time ''t'' flows like a [[fluid]]; hence the term ''probability current'', a [[vector field]]. The particle itself does ''not'' flow [[Deterministic system|deterministically]] in this [[vector field]].
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| :{| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
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| !Consistency with Schrödinger equation
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| |-
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| |For this derivation see<ref>Quantum Mechanics Demystified, D. McMahon, Mc Graw Hill (USA), 2006, ISBN(10) 0 07 145546 9</ref> for example. The 3-d time dependent [[Schrödinger equation]] and its [[complex conjugate]] (''i'' → –''i'' throughout) are respectively:
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| :<math> \begin{align} & -\frac{\hbar^2}{2m}\nabla^2 \Psi + U\Psi = i\hbar \frac{\partial \Psi}{\partial t} , \\
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| & - \frac{\hbar^2}{2m}\nabla^2 \Psi^{*} + U\Psi^{*} = - i\hbar \frac{\partial \Psi^{*}}{\partial t} ,\\
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| \end{align}</math>
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| where ''U'' is the [[Potential|potential function]]. The [[partial derivative]] of ''ρ'' with respect to ''t'' is:
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| :<math> \frac{\partial \rho}{\partial t} = \frac{\partial |\Psi |^2}{\partial t } = \frac{\partial}{\partial t} \left ( \Psi^{*} \Psi \right ) = \Psi^{*} \frac{\partial \Psi}{\partial t} + \Psi \frac{\partial\Psi^{*}}{\partial t} .</math>
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| Multiplying the Schrödinger equation by ''Ψ''* then solving for <math> \scriptstyle \Psi^{*} \partial \Psi/\partial t \,\!</math>, and similarly multiplying the complex conjugated Schrödinger equation by ''Ψ'' then solving for <math>\scriptstyle \Psi \partial \Psi^* / \partial t \,\!</math>;
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| :<math> \begin{align} & \Psi^*\frac{\partial \Psi}{\partial t} = \frac{1}{i\hbar } \left [ -\frac{\hbar^2\Psi^*}{2m}\nabla^2 \Psi + U\Psi^*\Psi \right ], \\
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| & \Psi \frac{\partial \Psi^*}{\partial t} = - \frac{1}{i\hbar } \left [ - \frac{\hbar^2\Psi}{2m}\nabla^2 \Psi^* + U\Psi\Psi^* \right ],\\
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| \end{align}</math>
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| substituting into the time derivative of ''ρ'':
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| :<math> \begin{align}
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| \frac{\partial \rho}{\partial t} & = \frac{1}{i\hbar } \left [ -\frac{\hbar^2\Psi^{*}}{2m}\nabla^2 \Psi + U\Psi^{*}\Psi \right ] - \frac{1}{i\hbar } \left [ - \frac{\hbar^2\Psi}{2m}\nabla^2 \Psi^{*} + U\Psi\Psi^{*} \right ] \\
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| & = \frac{1}{i\hbar } \left [ -\frac{\hbar^2\Psi^{*}}{2m}\nabla^2 \Psi + U\Psi^{*}\Psi \right ] + \frac{1}{i\hbar } \left [ +\frac{\hbar^2\Psi}{2m}\nabla^2 \Psi^{*} - U\Psi^{*}\Psi \right ] \\
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| & = - \frac{1}{i\hbar } \frac{\hbar^2\Psi^{*}}{2m}\nabla^2 \Psi + \frac{1}{i\hbar } \frac{\hbar^2\Psi}{2m}\nabla^2 \Psi^{*} \\
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| & = \frac{\hbar}{2im} \left [ \Psi\nabla^2 \Psi^{*} - \Psi^{*}\nabla^2 \Psi \right ] \\
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| \end{align} </math>
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| The [[Laplace operator|Laplacian]] [[Operator (mathematics)|operators]] (∇<sup>2</sup>) in the above result suggest that the right hand side is the divergence of '''j''', and the reversed order of terms imply this is the negative of '''j''', altogether:
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| :<math>\begin{align} \nabla \cdot \mathbf{j} & = \nabla \cdot \left [ \frac{\hbar}{2mi} \left ( \Psi^{*} \left ( \nabla \Psi \right ) - \Psi \left ( \nabla \Psi^{*} \right ) \right ) \right ] \\
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| & = \frac{\hbar}{2mi} \left [ \Psi^{*} \left ( \nabla^2 \Psi \right ) - \Psi \left ( \nabla^2 \Psi^{*} \right ) \right ] \\
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| & = - \frac{\hbar}{2mi} \left [ \Psi \left ( \nabla^2 \Psi^{*} \right ) - \Psi^{*} \left ( \nabla^2 \Psi \right ) \right ] \\
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| \end{align} </math>
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| so the continuity equation is:
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| :<math> \begin{align} & \frac{\partial \rho}{\partial t} = - \nabla \cdot \mathbf{j} \\
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| & \frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{j} = 0 \\
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| \end{align}</math>
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| The integral form follows as for the general equation.
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| |}
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| ==General relativity==
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| The [[stress-energy tensor]] is a second-order [[tensor field]] containing energy-momentum densities, energy-momentum fluxes, and shear stresses, of a mass-energy distribution. The [[Tensors in curvilinear coordinates#Second-order tensor field|divergence of this tensor]] vanishes:<ref>{{cite book|title=Gravitation|author=J.A. Wheeler, C. Misner, K.S. Thorne|publisher=W.H. Freeman & Co|year=1973|isbn=0-7167-0344-0}}</ref>
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| :<math>\frac{\partial T^{\alpha\beta}}{\partial x^\beta}=0</math>
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| because energy and momentum are conserved. This is an important constraint on the form the [[Einstein field equations]] take in [[general relativity]].<ref>{{cite book|title=Relativity DeMystified|author=D. McMahon|publisher=Mc Graw Hill (USA)|year=2006|isbn=0-07-145545-0}}</ref>
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| ==Quantum chromodynamics==
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| [[Quark]]s and [[gluon]]s have ''[[color charge]]'', which is much more complicated to describe mathematically than those described above. Nevertheless, color charge is conserved as well, so there is a continuity equation for color currents:
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| :<math>\partial_\nu j^\nu = 0</math>
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| See [[Gluon field strength tensor#Equation of motion|gluon field strength tensor]] for explicit expressions.
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| ==Conserved currents from Noether's theorem==
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| {{for|more detailed explanations and derivations|Noether's theorem}}
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| If a [[field (physics)|physical field]] described by a function of space and time, ''ϕ''('''r''', ''t''), is [[variational derivative|varied]] by a small amount;
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| :<math>\phi \rightarrow \phi + \delta\phi</math>
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| then [[Noether's theorem]] states the [[Lagrangian]] ''L'', or rather the [[Lagrangian density]] ℒ for [[field (physics)|field]]s, is invariant (does not change):
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| :<math>\mathcal{L} \rightarrow \mathcal{L} + \delta\mathcal{L}, \ \delta\mathcal{L}=0</math> | |
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| under a [[continuous symmetry]], in which the field is a continuous [[Variable (mathematics)|variable]]. The field {{nowrap|''ϕ''('''r''', ''t'')}} can be a [[scalar field]], often a [[scalar potential]] like the classical [[gravitational potential]] {{nowrap|''ζ''('''r''', ''t'')}} or [[electric potential]] {{nowrap|''ϕ''('''r''', ''t'')}}, or a [[vector field]] like the [[Newtonian gravitation|Newtonian gravitational]] field {{nowrap|'''g'''('''r''', ''t'')}} or [[vector potential]]s like the [[magnetic potential]] {{nowrap|'''A'''('''r''', ''t'')}}, or a [[tensor field]] of [[Mixed tensor|any order]] such as the [[electromagnetic tensor]] ''F''('''{{nowrap|r''', ''t'')}}, or even a [[spinor field]] which arise in describing particles as [[quantum field]]s. The Lagrangian density is a function of the field and its space and time derivatives: {{nowrap|ℒ[''ϕ''('''r''', ''t''), {{math|∂}}''ϕ''('''r''', ''t'')/{{math|∂}}''t'', ∇''ϕ''('''r''', ''t'')]}}.
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| The above considerations leads to conserved current densities in a completely general form:<ref>{{cite book| title=Quantum Field Theory|author=D. McMahon, Mc Graw Hill (USA)|year=2008|isbn=978-0-07-154382-8}}</ref>
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| :<math>J^\mu (\mathbf{r},t) = \frac{\partial \mathcal{L}}{\partial [\partial_\mu \phi(\mathbf{r},t)]}\delta\phi(\mathbf{r},t) </math>
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| because they satisfy the continuity equation
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| :<math>\partial_\mu J^\mu = 0</math>
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| :{| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
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| !Derivation of conserved currents from Lagrangian density
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| |-
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| The variation in ℒ is:
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| :<math>\begin{align}
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| \delta\mathcal{L} & = \frac{\partial \mathcal{L}}{\partial \phi}\delta\phi + \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)} \delta (\partial_\mu \phi)\\
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| & = \frac{\partial \mathcal{L}}{\partial \phi}\delta\phi + \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)} (\partial_\mu \delta \phi)\\
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| \end{align}</math>
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| next using the [[Euler-Lagrange equation]]s for a field:
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| :<math>\partial_\mu\left[\frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)}\right] = \frac{\partial \mathcal{L}}{\partial \phi}</math>
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| the variation is:
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| :<math>\begin{align}
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| \delta\mathcal{L} & = \partial_\mu\left[\frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)}\right]\delta\phi + \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)} (\partial_\mu \delta \phi)\\
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| & = \partial_\mu\left[\frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)}\delta\phi\right] \\
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| & = 0
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| \end{align}</math>
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| where the [[product rule]] has been used, so we obtain the continuity equation in a different general form:
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| :<math>\partial_\mu\left[\frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)}\delta\phi\right] = 0 </math>
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| So the conserved current density has to be:
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| :<math>J^\mu = \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)}\delta\phi </math>
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| |}
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| Integrating the density of the conserved quantity ''J''<sup>0</sup>/''c'', over a [[spacelike]] region of volume ''V'', gives the total amount of conserved quantity within that volume:
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| :<math>q(t) = \int_V \frac{1}{c} J^0(\mathbf{r},t) \, dV </math>
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| Equivalently, integrating the current density of the conserved quantity '''j''' = (''J''<sup>1</sup>, ''J''<sup>2</sup>, ''J''<sup>3</sup>), through a closed spacelike surface ''S'' (see above descriptions) over some time duration, also gives the conserved quantity flowing through that surface in that time duration:
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| :<math>q(t_2 - t_1) = \int_{t_1}^{t_2} \iint_S \mathbf{j}(\mathbf{r},t) \cdot d\mathbf{S} \,dt\, </math>
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| == See also ==
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| * [[Conservation law]]
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| * [[Groundwater energy balance]]
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| == References ==
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| {{reflist}}
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| ==Further reading==
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| *''Hydrodynamics, H. Lamb'', Cambridge University Press, (2006 digitalization of 1932 6th edition) ISBN 978-0-521-45868-9
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| *''Introduction to Electrodynamics (3rd Edition), D.J. Griffiths'', Pearson Education Inc, 1999, ISBN 81-7758-293-3
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| *''Electromagnetism (2nd edition), I.S. Grant, W.R. Phillips'', Manchester Physics Series, 2008 ISBN 0-471-92712-0
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| *''Gravitation, J.A. Wheeler, C. Misner, K.S. Thorne'', W.H. Freeman & Co, 1973, ISBN 0-7167-0344-0
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| [[Category:Equations of fluid dynamics]]
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| [[Category:Conservation equations]]
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| [[Category:Partial differential equations]]
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