5244:
1459:
4578:
1079:
3998:
36:
4211:
5239:{\displaystyle {\begin{aligned}{\frac {\partial \rho }{\partial t}}&={\frac {1}{i\hbar }}\left-{\frac {1}{i\hbar }}\left\\&={\frac {1}{i\hbar }}\left+{\frac {1}{i\hbar }}\left\\&=-{\frac {1}{i\hbar }}{\frac {\hbar ^{2}\Psi ^{*}}{2m}}\nabla ^{2}\Psi +{\frac {1}{i\hbar }}{\frac {\hbar ^{2}\Psi }{2m}}\nabla ^{2}\Psi ^{*}\\&={\frac {\hbar }{2im}}\left\\\end{aligned}}}
6763:
4565:
5744:
3784:
6228:
5954:
4029:
2853:
3714:
6387:
5628:
826:
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
2483:
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,
1714:
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
802:
states that energy can neither be created nor destroyedâi.e., the total amount of energy in the universe is fixed. This statement does not rule out the possibility that a quantity of energy could disappear from one point while simultaneously appearing at another point. A stronger statement is that
6380:
4270:
5633:
2307:
7949:
The laws of physics are invariant with respect to orientationâfor example, floating in outer space, there is no measurement you can do to say "which way is up"; the laws of physics are the same regardless of how you are oriented. This symmetry leads to the continuity equation for
7852:
6042:
5771:
7941:
The laws of physics are invariant with respect to space-translationâfor example, a rocket in outer space is not subject to different forces or potentials if it is displaced in any given direction (eg. x, y, z), leading to the conservation of the three components of momentum
1249:
2709:
7209:
3600:
7019:
3993:{\displaystyle {\begin{aligned}-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\Psi +U\Psi &=i\hbar {\frac {\partial \Psi }{\partial t}},\\-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\Psi ^{*}+U\Psi ^{*}&=-i\hbar {\frac {\partial \Psi ^{*}}{\partial t}},\\\end{aligned}}}
2687:
of the velocity field is zero everywhere. Physically, this is equivalent to saying that the local volume dilation rate is zero, hence a flow of water through a converging pipe will adjust solely by increasing its velocity as water is largely incompressible.
3320:
5274:
7442:
4206:{\displaystyle {\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}}.}
2378:
2211:
6333:
3458:
2517:, the continuity equation states that the rate at which mass enters a system is equal to the rate at which mass leaves the system plus the accumulation of mass within the system. The differential form of the continuity equation is:
2581:
2218:
1803:
954:
flowing per unit time, through a unit area". For example, in the mass continuity equation for flowing water, if 1 gram per second of water is flowing through a pipe with cross-sectional area 1 cm, then the average mass flux
6758:{\displaystyle {\begin{aligned}{\frac {dn}{dt}}A\,dx&=\left{\frac {A}{e}}+(G_{n}-R_{n})A\,dx\\&=\left{\frac {A}{e}}+(G_{n}-R_{n})A\,dx\\{\frac {dn}{dt}}&={\frac {1}{e}}{\frac {dJ}{dx}}+(G_{n}-R_{n})\end{aligned}}}
1453:
7743:
2478:
3115:. The flux in this case is the probability per unit area per unit time that the particle passes through a surface. According to the continuity equation, the negative divergence of this flux equals the rate of change of the
2706:, optical flow is the pattern of apparent motion of objects in a visual scene. Under the assumption that brightness of the moving object did not change between two image frames, one can derive the optical flow equation as:
2024:
2133:
3040:
3592:
7026:
4583:
6862:
6392:
7663:
containing energyâmomentum densities, energyâmomentum fluxes, and shear stresses, of a mass-energy distribution. The differential form of energyâmomentum conservation in general relativity states that the
4560:{\displaystyle {\begin{aligned}\Psi ^{*}{\frac {\partial \Psi }{\partial t}}&={\frac {1}{i\hbar }}\left,\\\Psi {\frac {\partial \Psi ^{*}}{\partial t}}&=-{\frac {1}{i\hbar }}\left,\\\end{aligned}}}
5739:{\displaystyle {\begin{aligned}&{\frac {\partial \rho }{\partial t}}=-\nabla \cdot \mathbf {j} \\{}\Rightarrow {}&{\frac {\partial \rho }{\partial t}}+\nabla \cdot \mathbf {j} =0\\\end{aligned}}}
5638:
1186:
7863:
form of the continuity equation is difficult to define and not necessarily valid for a region within which spacetime is significantly curved (e.g. around a black hole, or across the whole universe).
6855:
3195:
7348:
6769:
denotes current density(whose direction is against electron flow by convention) due to electron flow within the considered volume of the semiconductor. It is also called electron current density.
5279:
4275:
3789:
6223:{\displaystyle {\frac {\partial p}{\partial t}}=-p\mu _{p}{\frac {\partial E}{\partial x}}-\mu _{p}E{\frac {\partial p}{\partial x}}+D_{p}{\frac {\partial ^{2}p}{\partial x^{2}}}+(G_{p}-R_{p})}
3119:. The continuity equation reflects the fact that the molecule is always somewhereâthe integral of its probability distribution is always equal to 1âand that it moves by a continuous motion (no
5949:{\displaystyle {\frac {\partial n}{\partial t}}=n\mu _{n}{\frac {\partial E}{\partial x}}+\mu _{n}E{\frac {\partial n}{\partial x}}+D_{n}{\frac {\partial ^{2}n}{\partial x^{2}}}+(G_{n}-R_{n})}
7367:
2421:
7218:
The key to solving these equations in real devices is whenever possible to select regions in which most of the mechanisms are negligible so that the equations reduce to a much simpler form.
7621:
2312:
1035:
815:" from one place to anotherâit can only move by a continuous flow. A continuity equation is the mathematical way to express this kind of statement. For example, the continuity equation for
2681:
2848:{\displaystyle {\frac {\partial I}{\partial x}}V_{x}+{\frac {\partial I}{\partial y}}V_{y}+{\frac {\partial I}{\partial t}}=\nabla I\cdot \mathbf {V} +{\frac {\partial I}{\partial t}}=0}
2152:
193:
7565:
7515:
7714:
3709:{\displaystyle \nabla \cdot \mathbf {j} +{\frac {\partial \rho }{\partial t}}=0\mathrel {\rightleftharpoons } \nabla \cdot \mathbf {j} +{\frac {\partial |\Psi |^{2}}{\partial t}}=0.}
1546:
1288:
3339:
1690:
7517:
in the usual case where there are no sources or sinks, that is, for perfectly conserved quantities like energy or charge. This continuity equation is manifestly ("obviously")
2520:
2952:
791:
and other natural quantities are conserved under their respective appropriate conditions, a variety of physical phenomena may be described using continuity equations.
2491:
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.
6264:
5990:
2426:
5623:{\displaystyle {\begin{aligned}\nabla \cdot \mathbf {j} &=\nabla \cdot \left\\&={\frac {\hbar }{2mi}}\left\\&=-{\frac {\hbar }{2mi}}\left\\\end{aligned}}}
1972:
2084:
7884:, which is always conserved like electric charge, and there is a continuity equation for such color charge currents (explicit expressions for currents are given at
2988:
742:
3469:
1751:
6375:{\displaystyle {\text{Rate of change of electron density}}=({\text{Electron flux in}}-{\text{Electron flux out}})+{\text{Net generation inside a volume}}}
1386:
3138:. The terms in the equation require the following definitions, and are slightly less obvious than the other examples above, so they are outlined here:
1923:
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
8297:
3107:
If there is a quantity that moves continuously according to a stochastic (random) process, like the location of a single dissolved molecule with
827:
equation for the number of people alive; it has a "source term" to account for people being born, and a "sink term" to account for people dying.
2302:{\displaystyle \nabla \cdot (\nabla \times \mathbf {H} )=\nabla \cdot \mathbf {J} +{\frac {\partial (\nabla \cdot \mathbf {D} )}{\partial t}},}
6775:
965:, and its direction is along the pipe in the direction that the water is flowing. Outside the pipe, where there is no water, the flux is zero.
8247:
8209:
8016:
7275:
8287:
7645:, where spacetime is curved, the continuity equation (in differential form) for energy, charge, or other conserved quantities involves the
53:
7847:{\displaystyle \partial _{\mu }T^{\mu \nu }=-\Gamma _{\mu \lambda }^{\mu }T^{\lambda \nu }-\Gamma _{\mu \lambda }^{\nu }T^{\mu \lambda },}
1939:, have a similar mathematical form to the continuity equation, but are not usually referred to by the term "continuity equation", because
7733:
6315:
This section presents a derivation of the equation above for electrons. A similar derivation can be found for the equation for holes.
2389:
772:
842:
8268:
8228:
8180:
8134:
8109:
8084:
7580:
1244:{\displaystyle ({\text{Rate that }}q{\text{ is flowing through the imaginary surface }}S)=\iint _{S}\mathbf {j} \cdot d\mathbf {S} }
1007:
735:
119:
8004:
2649:
100:
7951:
6318:
Consider the fact that the number of electrons is conserved across a volume of semiconductor material with cross-sectional area,
2628:
7934:âfor example, the laws of physics today are the same as they were yesterday. This symmetry leads to the continuity equation for
7527:
7477:
7204:{\displaystyle {\frac {dn}{dt}}=\mu _{n}E{\frac {dn}{dx}}+\mu _{n}n{\frac {dE}{dx}}+D_{n}{\frac {d^{2}n}{dx^{2}}}+(G_{n}-R_{n})}
72:
8292:
7671:
708:
57:
409:
246:
7014:{\displaystyle {\frac {dn}{dt}}={\frac {1}{e}}{\frac {d}{dx}}\left(en\mu _{n}E+eD_{n}{\frac {dn}{dx}}\right)+(G_{n}-R_{n})}
79:
3189:
846:
728:
449:
335:
2632:
1924:
850:
404:
7885:
3158:
3154:
313:
196:
86:
7964:
3718:
Either form may be quoted. Intuitively, the above quantities indicate this represents the flow of probability. The
1955:
1936:
795:
1302:
in some literature, in which context "flux" denotes the surface integral of flux density. See the main article on
3315:{\displaystyle \rho (\mathbf {r} ,t)=\Psi ^{*}(\mathbf {r} ,t)\Psi (\mathbf {r} ,t)=|\Psi (\mathbf {r} ,t)|^{2}.}
320:
149:
7656:
7630:
2985:
for the nuances associated with general relativity.) Therefore, there is a continuity equation for energy flow:
68:
7943:
7719:
3112:
1071:
615:
610:
225:
399:
392:
7903:, and others. Each of these has a corresponding continuity equation, possibly including source / sink terms.
2627:
term represents the difference in flow in versus flow out. In this context, this equation is also one of the
1516:
3764:
678:
673:
342:
46:
7926:, there is a continuity equation for some conserved physical quantity. The three most famous examples are:
2646:(volumetric strain rate is zero), the mass continuity equation simplifies to a volume continuity equation:
7935:
5252:
2978:
2142:
2048:
2040:
799:
230:
7437:{\displaystyle \partial _{\mu }J^{\mu }=c{\frac {\partial \rho }{\partial ct}}+\nabla \cdot \mathbf {j} }
1649:
1258:
6285:
6014:
1058:
767:
that describes the transport of some quantity. It is particularly simple and powerful when applied to a
653:
271:
2373:{\displaystyle \nabla \cdot \mathbf {J} +{\frac {\partial (\nabla \cdot \mathbf {D} )}{\partial t}}=0.}
2215:
Taking the divergence of both sides (the divergence and partial derivative in time commute) results in
8151:
8046:
7919:
7912:
7646:
3749:
2623:
The time derivative can be understood as the accumulation (or loss) of mass in the system, while the
491:
308:
288:
276:
220:
2206:{\displaystyle \nabla \times \mathbf {H} =\mathbf {J} +{\frac {\partial \mathbf {D} }{\partial t}}.}
7923:
3463:
3116:
2911:
2643:
2044:
2034:
693:
541:
434:
140:
1458:
819:
states that the amount of electric charge in any volume of space can only change by the amount of
8259:
7974:
7723:
7642:
7524:
Examples of continuity equations often written in this form include electric charge conservation
7238:
4011:
3076:
2508:
1928:
1742:
768:
713:
347:
303:
298:
93:
2146:
2051:, although charge conservation is more fundamental than Maxwell's equations. It states that the
1368:
Mathematically, the integral form of the continuity equation expressing the rate of increase of
1488:
be a surface with boundaries, like those on the right. (Surfaces are blue, boundaries are red.)
8264:
8243:
8224:
8205:
8176:
8130:
8105:
8080:
8012:
7969:
7518:
6242:
5993:
5968:
5762:
3768:
3131:
2488:
1736:
1078:
330:
281:
8008:
7998:
7931:
6772:
Total electron current density is the sum of drift current and diffusion current densities:
6273:
6002:
5249:
3453:{\displaystyle P=P_{\mathbf {r} \in V}(t)=\int _{V}\Psi ^{*}\Psi dV=\int _{V}|\Psi |^{2}dV.}
1732:
1549:
1291:
1158:
820:
668:
643:
556:
531:
526:
481:
3108:
2703:
2636:
2056:
1067:
898:
816:
788:
658:
582:
496:
427:
361:
263:
7891:
There are many other quantities in particle physics which are often or always conserved:
546:
416:
7464:
7357:
3052:
2576:{\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot (\rho \mathbf {u} )=0}
2514:
2504:
2383:
2069:
1932:
1502:
1469:
969:
857:
663:
521:
486:
387:
293:
1798:{\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot \mathbf {j} =\sigma }
1692:
or the total change of q from its generation or destruction inside the control volume.
8281:
7896:
7892:
5758:
3120:
3096:
3080:
3072:
2613:
812:
703:
536:
1330:
flows inward through the surface of the region, and decreases when it flows outward;
834:), which applies to any finite region, or in a "differential form" (in terms of the
17:
7880:
7660:
3753:
3741:
3143:
2697:
2616:
2423:
which can be substituted in the previous equation to yield the continuity equation
937:
688:
683:
648:
380:
1448:{\displaystyle {\frac {dq}{dt}}+\oint _{S}\mathbf {j} \cdot d\mathbf {S} =\Sigma }
2473:{\displaystyle \nabla \cdot \mathbf {J} +{\frac {\partial \rho }{\partial t}}=0.}
3325:
3064:
2019:{\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot \mathbf {j} =0}
1143:
698:
601:
35:
3075:. When heat flows inside a solid, the continuity equation can be combined with
2128:{\displaystyle \nabla \cdot \mathbf {J} =-{\frac {\partial \rho }{\partial t}}}
7454:
7361:
7246:
2684:
2624:
2052:
1816:
1745:, a general continuity equation can also be written in a "differential form":
1000:âthen the flux is by definition equal to the density times the velocity field:
835:
830:
Any continuity equation can be expressed in an "integral form" (in terms of a
620:
516:
3067:(transfer of energy per unit cross-sectional area per unit time) as a vector,
7574:
7471:
7468:
7269:
5765:
of both the electrons in the conduction band and holes in the valence band.
5259:) in the above result suggest that the right hand side is the divergence of
4007:
3035:{\displaystyle {\frac {\partial u}{\partial t}}+\nabla \cdot \mathbf {q} =0}
2500:
841:
Continuity equations underlie more specific transport equations such as the
592:
587:
421:
4241:, and similarly multiplying the complex conjugated SchrĂśdinger equation by
1298:(Note that the concept that is here called "flux" is alternatively termed
7265:
7242:
7232:
3150:
3092:
977:
which describes the relevant flowâin other words, if all of the quantity
902:
784:
764:
571:
476:
456:
442:
3587:{\displaystyle \mathbf {j} (\mathbf {r} ,t)={\frac {\hbar }{2mi}}\left.}
1945:
in those cases does not represent the flow of a real physical quantity.
1715:
surface), increases when someone in the building gives birth (a source,
7900:
7895:(proportional to the number of quarks minus the number of antiquarks),
2593:
2078:
2065:
912:
325:
7918:
One reason that conservation equations frequently occur in physics is
3079:(heat flux is proportional to temperature gradient) to arrive at the
1959:
894:
780:
466:
1039:
For example, if in the mass continuity equation for flowing water,
7875:
7871:
3733:
3134:
is another domain where there is a continuity equation related to
1457:
1077:
370:
3083:. The equation of heat flow may also have source terms: Although
2635:
form a vector continuity equation describing the conservation of
2068:
per square meter) is equal to the negative rate of change of the
2043:, the continuity equation is an empirical law expressing (local)
856:
Flows governed by continuity equations can be visualized using a
7856:
The right-hand side strictly vanishes for a flat geometry only.
5265:, and the reversed order of terms imply this is the negative of
1303:
890:
874:
831:
776:
883:
can be defined. To define flux, first there must be a quantity
506:
29:
2484:
the continuity equation amounts to a conservation of charge.
1146:
that can be differentiated into infinitesimal area elements).
1865:
is the vector field describing the movement of the quantity
1719:), and decreases when someone in the building dies (a sink,
7249:, offer a convenient way to write any continuity equation.
7023:
Applying the product rule results in the final expression:
3091:
can be created from other types of energy, for example via
1708:
could be the number of people in the building. The surface
823:
flowing into or out of that volume through its boundaries.
1919:) are referred to as a "sources" and "sinks" respectively.
1314:
The integral form of the continuity equation states that:
6850:{\displaystyle J_{n}=en\mu _{n}E+eD_{n}{\frac {dn}{dx}}}
5757:
The total current flow in the semiconductor consists of
3645:
807:
conserved: energy can neither be created nor destroyed,
7922:. This states that whenever the laws of physics have a
7343:{\displaystyle J=\left(c\rho ,j_{x},j_{y},j_{z}\right)}
5748:
The integral form follows as for the general equation.
3597:
With these definitions the continuity equation reads:
2981:
says that energy cannot be created or destroyed. (See
1261:
7746:
7674:
7583:
7530:
7480:
7370:
7278:
7029:
6865:
6778:
6390:
6336:
6245:
6045:
5971:
5774:
5636:
5277:
4581:
4273:
4032:
3787:
3603:
3472:
3342:
3198:
2991:
2914:
2712:
2652:
2523:
2429:
2392:
2315:
2221:
2155:
2087:
1975:
1754:
1652:
1519:
1389:
1189:
1010:
152:
7911:
For more detailed explanations and derivations, see
1203: is flowing through the imaginary surface
2047:. Mathematically it is an automatic consequence of
1893:per unit volume per unit time. Terms that generate
1082:Illustration of how the fluxes, or flux densities,
1057:would be the mass flux, also known as the material
794:Continuity equations are a stronger, local form of
60:. Unsourced material may be challenged and removed.
8257:Wheeler, J. A.; Misner, C.; Thorne, K. S. (1973).
7930:The laws of physics are invariant with respect to
7846:
7708:
7615:
7559:
7509:
7436:
7342:
7203:
7013:
6849:
6757:
6374:
6258:
6222:
5984:
5948:
5738:
5622:
5238:
4559:
4205:
3992:
3708:
3586:
3452:
3314:
3034:
2946:
2847:
2675:
2575:
2472:
2415:
2372:
2301:
2205:
2127:
2018:
1797:
1684:
1560:is the total amount of the quantity in the volume
1540:
1447:
1282:
1243:
1029:
187:
27:Equation describing the transport of some quantity
1646:more negative. This term is sometimes written as
1462:In the integral form of the continuity equation,
1345:is created inside the region, and decreases when
944:. Here are some examples and properties of flux:
930:is flowing is described by its flux. The flux of
8035:, Section 3.3, Pitman Publishing Limited, London
7718:This is an important constraint on the form the
7668:divergence of the stress-energy tensor is zero:
2416:{\displaystyle \nabla \cdot \mathbf {D} =\rho ,}
8171:C.W. Misner; K.S. Thorne; J.A. Wheeler (1973).
8125:C.W. Misner; K.S. Thorne; J.A. Wheeler (1973).
7616:{\displaystyle \partial _{\nu }T^{\mu \nu }=0}
6384:Mathematically, this equality can be written:
3111:, then there is a continuity equation for its
2309:but the divergence of a curl is zero, so that
1030:{\displaystyle \mathbf {j} =\rho \mathbf {u} }
5768:General form for electrons in one-dimension:
1958:that cannot be created or destroyed (such as
736:
8:
8242:. Manchester Physics Series (2nd ed.).
8204:(6th ed.). Cambridge University Press.
8152:"Is Energy Conserved in General Relativity?"
2676:{\displaystyle \nabla \cdot \mathbf {u} =0,}
798:. For example, a weak version of the law of
771:, but it can be generalized to apply to any
8175:. W.H. Freeman & Co. pp. 558â559.
2884:is the image intensity at image coordinate
1051:is the water's density at each point, then
1045:is the water's velocity at each point, and
188:{\displaystyle J=-D{\frac {d\varphi }{dx}}}
7992:
7990:
7560:{\displaystyle \partial _{\mu }J^{\mu }=0}
7510:{\displaystyle \partial _{\mu }J^{\mu }=0}
743:
729:
576:
366:
209:
131:
7832:
7822:
7814:
7798:
7788:
7780:
7761:
7751:
7745:
7709:{\displaystyle {T^{\mu }}_{\nu ;\mu }=0.}
7688:
7681:
7676:
7673:
7598:
7588:
7582:
7545:
7535:
7529:
7495:
7485:
7479:
7429:
7397:
7385:
7375:
7369:
7329:
7316:
7303:
7277:
7192:
7179:
7160:
7142:
7135:
7129:
7102:
7093:
7066:
7057:
7030:
7028:
7002:
6989:
6954:
6948:
6929:
6899:
6889:
6866:
6864:
6827:
6821:
6802:
6783:
6777:
6742:
6729:
6699:
6689:
6662:
6651:
6639:
6626:
6606:
6560:
6523:
6511:
6498:
6478:
6418:
6395:
6391:
6389:
6367:
6356:
6348:
6337:
6335:
6250:
6244:
6211:
6198:
6179:
6161:
6154:
6148:
6121:
6112:
6085:
6079:
6046:
6044:
6035:is the rate of recombination of electrons
5976:
5970:
5937:
5924:
5905:
5887:
5880:
5874:
5847:
5838:
5811:
5805:
5775:
5773:
5721:
5692:
5688:
5683:
5674:
5642:
5637:
5635:
5597:
5582:
5564:
5554:
5519:
5490:
5480:
5451:
5436:
5409:
5378:
5338:
5311:
5288:
5278:
5276:
5218:
5208:
5195:
5185:
5155:
5139:
5129:
5105:
5098:
5083:
5071:
5050:
5040:
5033:
5018:
4991:
4975:
4965:
4941:
4934:
4911:
4894:
4875:
4854:
4844:
4837:
4814:
4793:
4774:
4764:
4740:
4733:
4710:
4693:
4674:
4653:
4643:
4636:
4613:
4586:
4582:
4580:
4569:substituting into the time derivative of
4539:
4520:
4510:
4486:
4479:
4456:
4429:
4419:
4395:
4376:
4355:
4345:
4338:
4315:
4288:
4282:
4274:
4272:
4183:
4173:
4147:
4141:
4120:
4096:
4076:
4071:
4062:
4056:
4033:
4031:
3966:
3956:
3934:
3918:
3908:
3888:
3882:
3852:
3821:
3801:
3795:
3788:
3786:
3722:of finding the particle at some position
3683:
3678:
3669:
3663:
3655:
3644:
3618:
3610:
3602:
3565:
3525:
3498:
3481:
3473:
3471:
3435:
3430:
3421:
3415:
3393:
3383:
3354:
3353:
3341:
3303:
3298:
3283:
3272:
3255:
3235:
3226:
3205:
3197:
3021:
2992:
2990:
2935:
2922:
2913:
2819:
2811:
2779:
2770:
2746:
2737:
2713:
2711:
2659:
2651:
2559:
2524:
2522:
2444:
2436:
2428:
2399:
2391:
2345:
2330:
2322:
2314:
2277:
2262:
2254:
2237:
2220:
2184:
2178:
2170:
2162:
2154:
2105:
2094:
2086:
2005:
1976:
1974:
1784:
1755:
1753:
1676:
1671:
1659:
1651:
1533:
1524:
1518:
1434:
1423:
1417:
1390:
1388:
1354:Apart from these two processes, there is
1275:
1266:
1260:
1236:
1225:
1219:
1201:
1193:
1188:
1022:
1011:
1009:
915:of this quantity, that is, the amount of
165:
151:
120:Learn how and when to remove this message
4215:Multiplying the SchrĂśdinger equation by
2386:(another Maxwell equation), states that
2147:Ampère's law (with Maxwell's correction)
1478:, like any of the surfaces on the left.
8223:(3rd ed.). Pearson Education Inc.
7986:
5963:is the local concentration of electrons
5521:
5411:
5313:
5157:
5102:
5092:
5037:
5027:
4938:
4920:
4841:
4823:
4737:
4719:
4640:
4622:
4483:
4465:
4342:
4324:
3953:
3885:
3849:
3798:
3500:
2982:
1541:{\displaystyle \oint _{S}d\mathbf {S} }
879:A continuity equation is useful when a
600:
555:
505:
465:
369:
238:
212:
139:
8238:Grant, I. S.; Phillips, W. R. (2008).
7897:electron number, mu number, tau number
6026:is the rate of generation of electrons
1931:. Other equations in physics, such as
1324:in a region increases when additional
6306:is the rate of recombination of holes
3760:Consistency with SchrĂśdinger equation
1927:. This equation also generalizes the
1600:is being generated inside the volume
1066:In a well-known example, the flux of
7:
8073:For this derivation see for example
7652:instead of the ordinary divergence.
6330:-axis. More precisely, one can say:
2908:is the optical flow velocity vector
2138:Consistency with Maxwell's equations
1283:{\textstyle \iint _{S}d\mathbf {S} }
1173:that is passing through the surface
948:The dimension of flux is "amount of
838:operator) which applies at a point.
58:adding citations to reliable sources
7577:; and energyâmomentum conservation
7474:. Then the continuity equation is:
6237:is the local concentration of holes
3161:), that is, a function of position
1685:{\displaystyle dq/dt|_{\text{gen}}}
1632:is being destroyed, it is called a
1612:is being generated, it is called a
7811:
7777:
7748:
7585:
7532:
7482:
7423:
7408:
7400:
7372:
6339:Rate of change of electron density
6297:is the rate of generation of holes
6172:
6158:
6132:
6124:
6096:
6088:
6057:
6049:
5898:
5884:
5858:
5850:
5822:
5814:
5786:
5778:
5715:
5703:
5695:
5668:
5653:
5645:
5603:
5594:
5579:
5561:
5551:
5542:
5487:
5477:
5468:
5457:
5448:
5433:
5375:
5371:
5363:
5352:
5349:
5335:
5300:
5282:
5224:
5215:
5205:
5192:
5182:
5178:
5136:
5126:
5111:
5077:
5068:
5047:
4997:
4988:
4972:
4962:
4947:
4900:
4891:
4881:
4872:
4851:
4790:
4786:
4771:
4761:
4746:
4699:
4690:
4680:
4671:
4650:
4597:
4589:
4536:
4532:
4517:
4507:
4492:
4437:
4426:
4422:
4416:
4401:
4392:
4382:
4373:
4352:
4299:
4294:
4291:
4279:
4191:
4180:
4176:
4170:
4158:
4153:
4150:
4138:
4126:
4117:
4102:
4098:
4084:
4067:
4059:
4044:
4036:
3974:
3963:
3959:
3931:
3915:
3905:
3863:
3858:
3855:
3836:
3827:
3818:
3691:
3674:
3666:
3649:
3629:
3621:
3604:
3562:
3558:
3550:
3539:
3536:
3522:
3426:
3399:
3390:
3277:
3249:
3223:
3071:An important practical example is
3015:
3003:
2995:
2830:
2822:
2802:
2790:
2782:
2757:
2749:
2724:
2716:
2653:
2547:
2535:
2527:
2455:
2447:
2430:
2393:
2355:
2339:
2333:
2316:
2287:
2271:
2265:
2248:
2231:
2222:
2191:
2181:
2156:
2116:
2108:
2088:
1999:
1987:
1979:
1778:
1766:
1758:
1442:
25:
7736:of the stressâenergy tensor does
6272:is the electric field across the
6001:is the electric field across the
1933:Gauss's law of the electric field
1151:If there is an imaginary surface
7952:conservation of angular momentum
7430:
5722:
5675:
5289:
3656:
3611:
3482:
3474:
3355:
3284:
3256:
3236:
3206:
3087:cannot be created or destroyed,
3022:
2812:
2660:
2629:Euler equations (fluid dynamics)
2560:
2437:
2400:
2346:
2323:
2278:
2255:
2238:
2185:
2171:
2163:
2095:
2006:
1785:
1534:
1435:
1424:
1276:
1237:
1226:
1023:
1012:
905:, number of molecules, etc. Let
889:which can flow or move, such as
34:
8221:Introduction to Electrodynamics
5630:so the continuity equation is:
3328:of finding the particle within
1339:in a region increases when new
45:needs additional citations for
8298:Partial differential equations
8047:"The Basics of Fluid Dynamics"
7198:
7172:
7008:
6982:
6748:
6722:
6645:
6619:
6598:
6592:
6554:
6548:
6517:
6491:
6470:
6464:
6455:
6440:
6369:Net generation inside a volume
6361:
6345:
6217:
6191:
5943:
5917:
5685:
4072:
4063:
3781:throughout) are respectively:
3679:
3670:
3492:
3478:
3431:
3422:
3373:
3367:
3299:
3294:
3280:
3273:
3266:
3252:
3246:
3232:
3216:
3202:
2941:
2915:
2564:
2553:
2350:
2336:
2282:
2268:
2242:
2228:
1672:
1209:
1190:
1:
8077:Quantum Mechanics Demystified
2947:{\displaystyle (V_{x},V_{y})}
1827:is the density of the amount
1472:that fully encloses a volume
843:convectionâdiffusion equation
3190:probability density function
847:Boltzmann transport equation
8288:Equations of fluid dynamics
7886:gluon field strength tensor
3744:. The particle itself does
3136:conservation of probability
1106:pass through open surfaces
924:The way that this quantity
8314:
8263:. W. H. Freeman & Co.
8150:Michael Weiss; John Baez.
8000:Geophysical fluid dynamics
7910:
7252:The density of a quantity
7237:The notation and tools of
7230:
3336:is denoted and defined by
2695:
2498:
2032:
1969:and the equations become:
1730:
1167:is equal to the amount of
872:
8219:Griffiths, D. J. (1999).
8129:. W.H. Freeman & Co.
7997:Pedlosky, Joseph (1987).
3103:Probability distributions
3055:(energy per unit volume),
2876:coordinates in the image,
1702:could be a building, and
1552:over that closed surface,
1505:, that encloses a volume
7944:conservation of momentum
7720:Einstein field equations
6259:{\displaystyle \mu _{p}}
5985:{\displaystyle \mu _{n}}
3113:probability distribution
1072:electric current density
989:is moving with velocity
247:ClausiusâDuhem (entropy)
197:Fick's laws of diffusion
7906:
7264:can be combined into a
2633:NavierâStokes equations
1937:Gauss's law for gravity
1925:NavierâStokes equations
1847:is the flux density of
851:NavierâStokes equations
405:NavierâStokes equations
343:Material failure theory
8293:Conservation equations
8102:Relativity DeMystified
7936:conservation of energy
7859:As a consequence, the
7848:
7710:
7617:
7561:
7511:
7438:
7344:
7205:
7015:
6851:
6759:
6376:
6260:
6224:
6039:Similarly, for holes:
5986:
5950:
5740:
5624:
5240:
4561:
4207:
3994:
3710:
3588:
3466:(probability flux) is
3454:
3316:
3036:
2979:Conservation of energy
2948:
2849:
2677:
2577:
2474:
2417:
2374:
2303:
2207:
2129:
2041:electromagnetic theory
2020:
1799:
1686:
1542:
1489:
1449:
1364:in a region to change.
1284:
1245:
1147:
1031:
800:conservation of energy
189:
8104:. McGraw Hill (USA).
7849:
7711:
7618:
7562:
7512:
7439:
7345:
7206:
7016:
6852:
6760:
6377:
6286:diffusion coefficient
6261:
6225:
6015:diffusion coefficient
5987:
5951:
5741:
5625:
5241:
4562:
4208:
3995:
3711:
3589:
3455:
3317:
3037:
2949:
2850:
2683:which means that the
2678:
2578:
2475:
2418:
2375:
2304:
2208:
2130:
2021:
1887:is the generation of
1800:
1696:In a simple example,
1687:
1594:is the net rate that
1543:
1461:
1450:
1285:
1246:
1081:
1032:
940:, which we denote as
400:Bernoulli's principle
393:Archimedes' principle
190:
69:"Continuity equation"
8075:McMahon, D. (2006).
8031:Clancy, L.J.(1975),
7744:
7740:necessarily vanish:
7672:
7657:stressâenergy tensor
7631:stressâenergy tensor
7581:
7528:
7478:
7368:
7364:of this current is:
7276:
7222:Relativistic version
7027:
6863:
6776:
6388:
6334:
6243:
6043:
5969:
5772:
5634:
5275:
4579:
4271:
4030:
3785:
3765:SchrĂśdinger equation
3601:
3470:
3340:
3196:
2989:
2954:at image coordinate
2912:
2710:
2650:
2521:
2427:
2390:
2313:
2219:
2153:
2085:
1973:
1752:
1650:
1626:more positive. When
1606:per unit time. When
1517:
1387:
1259:
1187:
1008:
492:Cohesion (chemistry)
314:Infinitesimal strain
150:
54:improve this article
18:Continuity equations
8100:D. McMahon (2006).
8045:Fielding, Suzanne.
7924:continuous symmetry
7827:
7793:
6859:Therefore, we have
3763:The time dependent
3738:probability current
3464:probability current
3117:probability density
2143:Maxwell's equations
2049:Maxwell's equations
2045:charge conservation
2035:Charge conservation
1833:(i.e. the quantity
961:inside the pipe is
757:continuity equation
410:Poiseuille equation
141:Continuum mechanics
135:Part of a series on
8200:Lamb, H. (2006) .
7975:Dissipative system
7844:
7810:
7776:
7724:general relativity
7706:
7659:is a second-order
7643:general relativity
7637:General relativity
7613:
7557:
7507:
7434:
7340:
7239:special relativity
7227:Special relativity
7201:
7011:
6847:
6755:
6753:
6372:
6256:
6220:
5982:
5946:
5736:
5734:
5620:
5618:
5236:
5234:
4557:
4555:
4203:
4012:partial derivative
4008:potential function
3990:
3988:
3761:
3706:
3584:
3450:
3312:
3032:
2944:
2845:
2673:
2573:
2509:Vorticity equation
2489:magnetic monopoles
2470:
2413:
2370:
2299:
2203:
2139:
2125:
2081:per cubic meter),
2016:
1956:conserved quantity
1929:advection equation
1795:
1743:divergence theorem
1682:
1538:
1490:
1445:
1358:for the amount of
1280:
1241:
1148:
1027:
869:Definition of flux
773:extensive quantity
769:conserved quantity
761:transport equation
616:Magnetorheological
611:Electrorheological
348:Fracture mechanics
185:
8249:978-0-471-92712-9
8211:978-0-521-45868-9
8054:Durham University
8018:978-0-387-96387-7
7970:Conservation form
7920:Noether's theorem
7913:Noether's theorem
7907:Noether's theorem
7655:For example, the
7519:Lorentz invariant
7418:
7167:
7120:
7084:
7048:
6972:
6912:
6897:
6884:
6845:
6717:
6697:
6680:
6614:
6578:
6486:
6413:
6370:
6359:
6358:Electron flux out
6351:
6340:
6186:
6139:
6103:
6064:
5994:electron mobility
5912:
5865:
5829:
5793:
5763:diffusion current
5710:
5660:
5535:
5425:
5327:
5171:
5123:
5096:
5065:
5031:
4959:
4924:
4869:
4827:
4758:
4723:
4668:
4626:
4604:
4504:
4469:
4444:
4370:
4328:
4306:
4245:then solving for
4219:then solving for
4198:
4165:
4109:
4091:
4051:
3981:
3902:
3870:
3815:
3769:complex conjugate
3759:
3750:deterministically
3736:; hence the term
3698:
3636:
3514:
3132:Quantum mechanics
3127:Quantum mechanics
3010:
2837:
2797:
2764:
2731:
2542:
2462:
2362:
2294:
2198:
2137:
2123:
1994:
1948:In the case that
1839:per unit volume),
1773:
1737:conservation form
1727:Differential form
1679:
1501:is any imaginary
1408:
1204:
1196:
921:per unit volume.
796:conservation laws
753:
752:
628:
627:
562:
561:
331:Contact mechanics
254:
253:
183:
130:
129:
122:
104:
16:(Redirected from
8305:
8274:
8253:
8240:Electromagnetism
8234:
8215:
8187:
8186:
8168:
8162:
8161:
8159:
8158:
8147:
8141:
8140:
8122:
8116:
8115:
8097:
8091:
8090:
8071:
8065:
8064:
8062:
8060:
8051:
8042:
8036:
8029:
8023:
8022:
7994:
7965:Conservation law
7932:time-translation
7867:Particle physics
7853:
7851:
7850:
7845:
7840:
7839:
7826:
7821:
7806:
7805:
7792:
7787:
7769:
7768:
7756:
7755:
7715:
7713:
7712:
7707:
7699:
7698:
7687:
7686:
7685:
7628:
7622:
7620:
7619:
7614:
7606:
7605:
7593:
7592:
7573:is the electric
7572:
7566:
7564:
7563:
7558:
7550:
7549:
7540:
7539:
7516:
7514:
7513:
7508:
7500:
7499:
7490:
7489:
7462:
7452:
7443:
7441:
7440:
7435:
7433:
7419:
7417:
7406:
7398:
7390:
7389:
7380:
7379:
7355:
7349:
7347:
7346:
7341:
7339:
7335:
7334:
7333:
7321:
7320:
7308:
7307:
7263:
7258:and its current
7257:
7210:
7208:
7207:
7202:
7197:
7196:
7184:
7183:
7168:
7166:
7165:
7164:
7151:
7147:
7146:
7136:
7134:
7133:
7121:
7119:
7111:
7103:
7098:
7097:
7085:
7083:
7075:
7067:
7062:
7061:
7049:
7047:
7039:
7031:
7020:
7018:
7017:
7012:
7007:
7006:
6994:
6993:
6978:
6974:
6973:
6971:
6963:
6955:
6953:
6952:
6934:
6933:
6913:
6911:
6900:
6898:
6890:
6885:
6883:
6875:
6867:
6856:
6854:
6853:
6848:
6846:
6844:
6836:
6828:
6826:
6825:
6807:
6806:
6788:
6787:
6764:
6762:
6761:
6756:
6754:
6747:
6746:
6734:
6733:
6718:
6716:
6708:
6700:
6698:
6690:
6681:
6679:
6671:
6663:
6644:
6643:
6631:
6630:
6615:
6607:
6605:
6601:
6579:
6577:
6569:
6561:
6533:
6516:
6515:
6503:
6502:
6487:
6479:
6477:
6473:
6414:
6412:
6404:
6396:
6381:
6379:
6378:
6373:
6371:
6368:
6360:
6357:
6352:
6350:Electron flux in
6349:
6341:
6338:
6274:depletion region
6266:is hole mobility
6265:
6263:
6262:
6257:
6255:
6254:
6229:
6227:
6226:
6221:
6216:
6215:
6203:
6202:
6187:
6185:
6184:
6183:
6170:
6166:
6165:
6155:
6153:
6152:
6140:
6138:
6130:
6122:
6117:
6116:
6104:
6102:
6094:
6086:
6084:
6083:
6065:
6063:
6055:
6047:
6003:depletion region
5991:
5989:
5988:
5983:
5981:
5980:
5955:
5953:
5952:
5947:
5942:
5941:
5929:
5928:
5913:
5911:
5910:
5909:
5896:
5892:
5891:
5881:
5879:
5878:
5866:
5864:
5856:
5848:
5843:
5842:
5830:
5828:
5820:
5812:
5810:
5809:
5794:
5792:
5784:
5776:
5745:
5743:
5742:
5737:
5735:
5725:
5711:
5709:
5701:
5693:
5689:
5684:
5678:
5661:
5659:
5651:
5643:
5640:
5629:
5627:
5626:
5621:
5619:
5615:
5611:
5610:
5606:
5602:
5601:
5587:
5586:
5574:
5570:
5569:
5568:
5559:
5558:
5536:
5534:
5520:
5509:
5505:
5501:
5500:
5496:
5495:
5494:
5485:
5484:
5464:
5460:
5456:
5455:
5441:
5440:
5426:
5424:
5410:
5402:
5398:
5394:
5393:
5389:
5388:
5384:
5383:
5382:
5359:
5355:
5343:
5342:
5328:
5326:
5312:
5292:
5270:
5264:
5258:
5245:
5243:
5242:
5237:
5235:
5231:
5227:
5223:
5222:
5213:
5212:
5200:
5199:
5190:
5189:
5172:
5170:
5156:
5148:
5144:
5143:
5134:
5133:
5124:
5122:
5114:
5110:
5109:
5099:
5097:
5095:
5084:
5076:
5075:
5066:
5064:
5056:
5055:
5054:
5045:
5044:
5034:
5032:
5030:
5019:
5008:
5004:
5000:
4996:
4995:
4980:
4979:
4970:
4969:
4960:
4958:
4950:
4946:
4945:
4935:
4925:
4923:
4912:
4907:
4903:
4899:
4898:
4880:
4879:
4870:
4868:
4860:
4859:
4858:
4849:
4848:
4838:
4828:
4826:
4815:
4807:
4803:
4799:
4798:
4797:
4779:
4778:
4769:
4768:
4759:
4757:
4749:
4745:
4744:
4734:
4724:
4722:
4711:
4706:
4702:
4698:
4697:
4679:
4678:
4669:
4667:
4659:
4658:
4657:
4648:
4647:
4637:
4627:
4625:
4614:
4605:
4603:
4595:
4587:
4574:
4566:
4564:
4563:
4558:
4556:
4549:
4545:
4544:
4543:
4525:
4524:
4515:
4514:
4505:
4503:
4495:
4491:
4490:
4480:
4470:
4468:
4457:
4445:
4443:
4435:
4434:
4433:
4420:
4408:
4404:
4400:
4399:
4381:
4380:
4371:
4369:
4361:
4360:
4359:
4350:
4349:
4339:
4329:
4327:
4316:
4307:
4305:
4297:
4289:
4287:
4286:
4266:
4265:
4263:
4262:
4256:
4253:
4244:
4240:
4239:
4237:
4236:
4230:
4227:
4218:
4212:
4210:
4209:
4204:
4199:
4197:
4189:
4188:
4187:
4174:
4166:
4164:
4156:
4148:
4146:
4145:
4133:
4129:
4125:
4124:
4110:
4108:
4097:
4092:
4090:
4082:
4081:
4080:
4075:
4066:
4057:
4052:
4050:
4042:
4034:
4025:
4020:with respect to
4019:
4005:
3999:
3997:
3996:
3991:
3989:
3982:
3980:
3972:
3971:
3970:
3957:
3939:
3938:
3923:
3922:
3913:
3912:
3903:
3901:
3893:
3892:
3883:
3871:
3869:
3861:
3853:
3826:
3825:
3816:
3814:
3806:
3805:
3796:
3780:
3731:
3727:
3715:
3713:
3712:
3707:
3699:
3697:
3689:
3688:
3687:
3682:
3673:
3664:
3659:
3648:
3637:
3635:
3627:
3619:
3614:
3593:
3591:
3590:
3585:
3580:
3576:
3575:
3571:
3570:
3569:
3546:
3542:
3530:
3529:
3515:
3513:
3499:
3485:
3477:
3459:
3457:
3456:
3451:
3440:
3439:
3434:
3425:
3420:
3419:
3398:
3397:
3388:
3387:
3366:
3365:
3358:
3335:
3331:
3321:
3319:
3318:
3313:
3308:
3307:
3302:
3287:
3276:
3259:
3239:
3231:
3230:
3209:
3184:
3172:
3166:
3148:
3073:the flow of heat
3062:
3050:
3041:
3039:
3038:
3033:
3025:
3011:
3009:
3001:
2993:
2969:
2965:
2953:
2951:
2950:
2945:
2940:
2939:
2927:
2926:
2907:
2899:
2895:
2883:
2875:
2863:
2854:
2852:
2851:
2846:
2838:
2836:
2828:
2820:
2815:
2798:
2796:
2788:
2780:
2775:
2774:
2765:
2763:
2755:
2747:
2742:
2741:
2732:
2730:
2722:
2714:
2682:
2680:
2679:
2674:
2663:
2642:If the fluid is
2611:
2603:
2591:
2582:
2580:
2579:
2574:
2563:
2543:
2541:
2533:
2525:
2479:
2477:
2476:
2471:
2463:
2461:
2453:
2445:
2440:
2422:
2420:
2419:
2414:
2403:
2379:
2377:
2376:
2371:
2363:
2361:
2353:
2349:
2331:
2326:
2308:
2306:
2305:
2300:
2295:
2293:
2285:
2281:
2263:
2258:
2241:
2212:
2210:
2209:
2204:
2199:
2197:
2189:
2188:
2179:
2174:
2166:
2134:
2132:
2131:
2126:
2124:
2122:
2114:
2106:
2098:
2076:
2063:
2029:Electromagnetism
2025:
2023:
2022:
2017:
2009:
1995:
1993:
1985:
1977:
1968:
1953:
1944:
1918:
1911:
1905:
1898:
1892:
1886:
1878:
1870:
1852:
1846:
1838:
1832:
1826:
1814:
1804:
1802:
1801:
1796:
1788:
1774:
1772:
1764:
1756:
1733:Conservation law
1722:
1718:
1713:
1707:
1701:
1691:
1689:
1688:
1683:
1681:
1680:
1677:
1675:
1663:
1645:
1641:
1631:
1625:
1621:
1611:
1605:
1599:
1593:
1587:
1579:
1573:
1565:
1559:
1550:surface integral
1547:
1545:
1544:
1539:
1537:
1529:
1528:
1510:
1500:
1483:
1477:
1467:
1454:
1452:
1451:
1446:
1438:
1427:
1422:
1421:
1409:
1407:
1399:
1391:
1379:
1374:within a volume
1373:
1363:
1350:
1344:
1338:
1329:
1323:
1292:surface integral
1289:
1287:
1286:
1281:
1279:
1271:
1270:
1250:
1248:
1247:
1242:
1240:
1229:
1224:
1223:
1205:
1202:
1197:
1194:
1178:
1172:
1166:
1159:surface integral
1156:
1141:
1132:
1123:
1114:
1105:
1099:
1090:
1056:
1050:
1044:
1036:
1034:
1033:
1028:
1026:
1015:
999:
988:
982:
976:
964:
960:
953:
935:
929:
920:
910:
888:
864:General equation
821:electric current
745:
738:
731:
577:
542:Gay-Lussac's law
532:Combined gas law
482:Capillary action
367:
210:
194:
192:
191:
186:
184:
182:
174:
166:
132:
125:
118:
114:
111:
105:
103:
62:
38:
30:
21:
8313:
8312:
8308:
8307:
8306:
8304:
8303:
8302:
8278:
8277:
8271:
8256:
8250:
8237:
8231:
8218:
8212:
8199:
8196:
8194:Further reading
8191:
8190:
8183:
8170:
8169:
8165:
8156:
8154:
8149:
8148:
8144:
8137:
8124:
8123:
8119:
8112:
8099:
8098:
8094:
8087:
8079:. McGraw Hill.
8074:
8072:
8068:
8058:
8056:
8049:
8044:
8043:
8039:
8030:
8026:
8019:
7996:
7995:
7988:
7983:
7961:
7916:
7909:
7869:
7828:
7794:
7757:
7747:
7742:
7741:
7677:
7675:
7670:
7669:
7639:
7624:
7594:
7584:
7579:
7578:
7568:
7541:
7531:
7526:
7525:
7491:
7481:
7476:
7475:
7458:
7451:
7445:
7407:
7399:
7381:
7371:
7366:
7365:
7351:
7325:
7312:
7299:
7289:
7285:
7274:
7273:
7259:
7253:
7235:
7229:
7224:
7216:
7188:
7175:
7156:
7152:
7138:
7137:
7125:
7112:
7104:
7089:
7076:
7068:
7053:
7040:
7032:
7025:
7024:
6998:
6985:
6964:
6956:
6944:
6925:
6918:
6914:
6904:
6876:
6868:
6861:
6860:
6837:
6829:
6817:
6798:
6779:
6774:
6773:
6752:
6751:
6738:
6725:
6709:
6701:
6682:
6672:
6664:
6659:
6658:
6635:
6622:
6570:
6562:
6544:
6540:
6531:
6530:
6507:
6494:
6436:
6432:
6425:
6405:
6397:
6386:
6385:
6332:
6331:
6313:
6304:
6295:
6282:
6246:
6241:
6240:
6207:
6194:
6175:
6171:
6157:
6156:
6144:
6131:
6123:
6108:
6095:
6087:
6075:
6056:
6048:
6041:
6040:
6033:
6024:
6011:
5972:
5967:
5966:
5933:
5920:
5901:
5897:
5883:
5882:
5870:
5857:
5849:
5834:
5821:
5813:
5801:
5785:
5777:
5770:
5769:
5755:
5750:
5733:
5732:
5702:
5694:
5690:
5680:
5679:
5652:
5644:
5632:
5631:
5617:
5616:
5593:
5592:
5588:
5578:
5560:
5550:
5549:
5545:
5541:
5537:
5524:
5507:
5506:
5486:
5476:
5475:
5471:
5447:
5446:
5442:
5432:
5431:
5427:
5414:
5400:
5399:
5374:
5370:
5366:
5348:
5344:
5334:
5333:
5329:
5316:
5310:
5306:
5293:
5273:
5272:
5266:
5260:
5256:
5233:
5232:
5214:
5204:
5191:
5181:
5177:
5173:
5160:
5146:
5145:
5135:
5125:
5115:
5101:
5100:
5088:
5067:
5057:
5046:
5036:
5035:
5023:
5006:
5005:
4987:
4971:
4961:
4951:
4937:
4936:
4930:
4926:
4916:
4890:
4871:
4861:
4850:
4840:
4839:
4833:
4829:
4819:
4805:
4804:
4789:
4770:
4760:
4750:
4736:
4735:
4729:
4725:
4715:
4689:
4670:
4660:
4649:
4639:
4638:
4632:
4628:
4618:
4606:
4596:
4588:
4577:
4576:
4570:
4554:
4553:
4535:
4516:
4506:
4496:
4482:
4481:
4475:
4471:
4461:
4446:
4436:
4425:
4421:
4413:
4412:
4391:
4372:
4362:
4351:
4341:
4340:
4334:
4330:
4320:
4308:
4298:
4290:
4278:
4269:
4268:
4257:
4254:
4251:
4250:
4248:
4246:
4242:
4231:
4228:
4225:
4224:
4222:
4220:
4216:
4190:
4179:
4175:
4157:
4149:
4137:
4116:
4115:
4111:
4101:
4083:
4070:
4058:
4043:
4035:
4028:
4027:
4021:
4015:
4001:
3987:
3986:
3973:
3962:
3958:
3940:
3930:
3914:
3904:
3894:
3884:
3876:
3875:
3862:
3854:
3839:
3817:
3807:
3797:
3783:
3782:
3772:
3729:
3723:
3690:
3677:
3665:
3628:
3620:
3599:
3598:
3561:
3557:
3553:
3535:
3531:
3521:
3520:
3516:
3503:
3468:
3467:
3429:
3411:
3389:
3379:
3349:
3338:
3337:
3333:
3329:
3297:
3222:
3194:
3193:
3174:
3168:
3162:
3146:
3129:
3109:Brownian motion
3105:
3058:
3046:
3002:
2994:
2987:
2986:
2976:
2974:Energy and heat
2967:
2955:
2931:
2918:
2910:
2909:
2903:
2897:
2885:
2879:
2867:
2859:
2829:
2821:
2789:
2781:
2766:
2756:
2748:
2733:
2723:
2715:
2708:
2707:
2704:computer vision
2700:
2694:
2692:Computer vision
2648:
2647:
2637:linear momentum
2607:
2599:
2587:
2534:
2526:
2519:
2518:
2511:
2497:
2481:
2454:
2446:
2425:
2424:
2388:
2387:
2354:
2332:
2311:
2310:
2286:
2264:
2217:
2216:
2190:
2180:
2151:
2150:
2115:
2107:
2083:
2082:
2072:
2059:
2057:current density
2037:
2031:
1986:
1978:
1971:
1970:
1963:
1949:
1940:
1913:
1907:
1900:
1894:
1888:
1882:
1874:
1866:
1848:
1842:
1834:
1828:
1822:
1812:
1806:
1765:
1757:
1750:
1749:
1739:
1729:
1720:
1716:
1709:
1703:
1697:
1670:
1648:
1647:
1643:
1642:, and it makes
1637:
1627:
1623:
1622:, and it makes
1617:
1607:
1601:
1595:
1591:
1583:
1575:
1574:is the flux of
1569:
1561:
1555:
1520:
1515:
1514:
1506:
1496:
1479:
1473:
1463:
1456:
1413:
1400:
1392:
1385:
1384:
1375:
1369:
1359:
1346:
1340:
1334:
1325:
1319:
1312:
1262:
1257:
1256:
1252:
1215:
1195:Rate that
1185:
1184:
1174:
1168:
1162:
1152:
1140:
1134:
1131:
1125:
1122:
1116:
1113:
1107:
1101:
1098:
1092:
1089:
1083:
1068:electric charge
1052:
1046:
1040:
1006:
1005:
990:
984:
978:
972:
962:
956:
949:
931:
925:
916:
906:
899:electric charge
884:
877:
871:
866:
817:electric charge
789:electric charge
749:
720:
719:
718:
638:
630:
629:
583:Viscoelasticity
574:
564:
563:
551:
501:
497:Surface tension
461:
364:
362:Fluid mechanics
354:
353:
352:
266:
264:Solid mechanics
256:
255:
207:
199:
175:
167:
148:
147:
126:
115:
109:
106:
63:
61:
51:
39:
28:
23:
22:
15:
12:
11:
5:
8311:
8309:
8301:
8300:
8295:
8290:
8280:
8279:
8276:
8275:
8269:
8254:
8248:
8235:
8229:
8216:
8210:
8195:
8192:
8189:
8188:
8181:
8163:
8142:
8135:
8117:
8110:
8092:
8085:
8066:
8037:
8024:
8017:
7985:
7984:
7982:
7979:
7978:
7977:
7972:
7967:
7960:
7957:
7956:
7955:
7947:
7939:
7908:
7905:
7868:
7865:
7843:
7838:
7835:
7831:
7825:
7820:
7817:
7813:
7809:
7804:
7801:
7797:
7791:
7786:
7783:
7779:
7775:
7772:
7767:
7764:
7760:
7754:
7750:
7705:
7702:
7697:
7694:
7691:
7684:
7680:
7638:
7635:
7612:
7609:
7604:
7601:
7597:
7591:
7587:
7556:
7553:
7548:
7544:
7538:
7534:
7506:
7503:
7498:
7494:
7488:
7484:
7447:
7432:
7428:
7425:
7422:
7416:
7413:
7410:
7405:
7402:
7396:
7393:
7388:
7384:
7378:
7374:
7358:speed of light
7338:
7332:
7328:
7324:
7319:
7315:
7311:
7306:
7302:
7298:
7295:
7292:
7288:
7284:
7281:
7228:
7225:
7223:
7220:
7215:
7212:
7200:
7195:
7191:
7187:
7182:
7178:
7174:
7171:
7163:
7159:
7155:
7150:
7145:
7141:
7132:
7128:
7124:
7118:
7115:
7110:
7107:
7101:
7096:
7092:
7088:
7082:
7079:
7074:
7071:
7065:
7060:
7056:
7052:
7046:
7043:
7038:
7035:
7010:
7005:
7001:
6997:
6992:
6988:
6984:
6981:
6977:
6970:
6967:
6962:
6959:
6951:
6947:
6943:
6940:
6937:
6932:
6928:
6924:
6921:
6917:
6910:
6907:
6903:
6896:
6893:
6888:
6882:
6879:
6874:
6871:
6843:
6840:
6835:
6832:
6824:
6820:
6816:
6813:
6810:
6805:
6801:
6797:
6794:
6791:
6786:
6782:
6750:
6745:
6741:
6737:
6732:
6728:
6724:
6721:
6715:
6712:
6707:
6704:
6696:
6693:
6688:
6685:
6683:
6678:
6675:
6670:
6667:
6661:
6660:
6657:
6654:
6650:
6647:
6642:
6638:
6634:
6629:
6625:
6621:
6618:
6613:
6610:
6604:
6600:
6597:
6594:
6591:
6588:
6585:
6582:
6576:
6573:
6568:
6565:
6559:
6556:
6553:
6550:
6547:
6543:
6539:
6536:
6534:
6532:
6529:
6526:
6522:
6519:
6514:
6510:
6506:
6501:
6497:
6493:
6490:
6485:
6482:
6476:
6472:
6469:
6466:
6463:
6460:
6457:
6454:
6451:
6448:
6445:
6442:
6439:
6435:
6431:
6428:
6426:
6424:
6421:
6417:
6411:
6408:
6403:
6400:
6394:
6393:
6366:
6363:
6355:
6347:
6344:
6322:, and length,
6312:
6309:
6308:
6307:
6302:
6298:
6293:
6289:
6280:
6276:
6267:
6253:
6249:
6238:
6219:
6214:
6210:
6206:
6201:
6197:
6193:
6190:
6182:
6178:
6174:
6169:
6164:
6160:
6151:
6147:
6143:
6137:
6134:
6129:
6126:
6120:
6115:
6111:
6107:
6101:
6098:
6093:
6090:
6082:
6078:
6074:
6071:
6068:
6062:
6059:
6054:
6051:
6037:
6036:
6031:
6027:
6022:
6018:
6009:
6005:
5996:
5979:
5975:
5964:
5945:
5940:
5936:
5932:
5927:
5923:
5919:
5916:
5908:
5904:
5900:
5895:
5890:
5886:
5877:
5873:
5869:
5863:
5860:
5855:
5852:
5846:
5841:
5837:
5833:
5827:
5824:
5819:
5816:
5808:
5804:
5800:
5797:
5791:
5788:
5783:
5780:
5754:
5751:
5731:
5728:
5724:
5720:
5717:
5714:
5708:
5705:
5700:
5697:
5691:
5687:
5682:
5681:
5677:
5673:
5670:
5667:
5664:
5658:
5655:
5650:
5647:
5641:
5639:
5614:
5609:
5605:
5600:
5596:
5591:
5585:
5581:
5577:
5573:
5567:
5563:
5557:
5553:
5548:
5544:
5540:
5533:
5530:
5527:
5523:
5518:
5515:
5512:
5510:
5508:
5504:
5499:
5493:
5489:
5483:
5479:
5474:
5470:
5467:
5463:
5459:
5454:
5450:
5445:
5439:
5435:
5430:
5423:
5420:
5417:
5413:
5408:
5405:
5403:
5401:
5397:
5392:
5387:
5381:
5377:
5373:
5369:
5365:
5362:
5358:
5354:
5351:
5347:
5341:
5337:
5332:
5325:
5322:
5319:
5315:
5309:
5305:
5302:
5299:
5296:
5294:
5291:
5287:
5284:
5281:
5280:
5271:, altogether:
5230:
5226:
5221:
5217:
5211:
5207:
5203:
5198:
5194:
5188:
5184:
5180:
5176:
5169:
5166:
5163:
5159:
5154:
5151:
5149:
5147:
5142:
5138:
5132:
5128:
5121:
5118:
5113:
5108:
5104:
5094:
5091:
5087:
5082:
5079:
5074:
5070:
5063:
5060:
5053:
5049:
5043:
5039:
5029:
5026:
5022:
5017:
5014:
5011:
5009:
5007:
5003:
4999:
4994:
4990:
4986:
4983:
4978:
4974:
4968:
4964:
4957:
4954:
4949:
4944:
4940:
4933:
4929:
4922:
4919:
4915:
4910:
4906:
4902:
4897:
4893:
4889:
4886:
4883:
4878:
4874:
4867:
4864:
4857:
4853:
4847:
4843:
4836:
4832:
4825:
4822:
4818:
4813:
4810:
4808:
4806:
4802:
4796:
4792:
4788:
4785:
4782:
4777:
4773:
4767:
4763:
4756:
4753:
4748:
4743:
4739:
4732:
4728:
4721:
4718:
4714:
4709:
4705:
4701:
4696:
4692:
4688:
4685:
4682:
4677:
4673:
4666:
4663:
4656:
4652:
4646:
4642:
4635:
4631:
4624:
4621:
4617:
4612:
4609:
4607:
4602:
4599:
4594:
4591:
4585:
4584:
4552:
4548:
4542:
4538:
4534:
4531:
4528:
4523:
4519:
4513:
4509:
4502:
4499:
4494:
4489:
4485:
4478:
4474:
4467:
4464:
4460:
4455:
4452:
4449:
4447:
4442:
4439:
4432:
4428:
4424:
4418:
4415:
4414:
4411:
4407:
4403:
4398:
4394:
4390:
4387:
4384:
4379:
4375:
4368:
4365:
4358:
4354:
4348:
4344:
4337:
4333:
4326:
4323:
4319:
4314:
4311:
4309:
4304:
4301:
4296:
4293:
4285:
4281:
4277:
4276:
4202:
4196:
4193:
4186:
4182:
4178:
4172:
4169:
4163:
4160:
4155:
4152:
4144:
4140:
4136:
4132:
4128:
4123:
4119:
4114:
4107:
4104:
4100:
4095:
4089:
4086:
4079:
4074:
4069:
4065:
4061:
4055:
4049:
4046:
4041:
4038:
3985:
3979:
3976:
3969:
3965:
3961:
3955:
3952:
3949:
3946:
3943:
3941:
3937:
3933:
3929:
3926:
3921:
3917:
3911:
3907:
3900:
3897:
3891:
3887:
3881:
3878:
3877:
3874:
3868:
3865:
3860:
3857:
3851:
3848:
3845:
3842:
3840:
3838:
3835:
3832:
3829:
3824:
3820:
3813:
3810:
3804:
3800:
3794:
3791:
3790:
3758:
3705:
3702:
3696:
3693:
3686:
3681:
3676:
3672:
3668:
3662:
3658:
3654:
3651:
3647:
3643:
3640:
3634:
3631:
3626:
3623:
3617:
3613:
3609:
3606:
3595:
3594:
3583:
3579:
3574:
3568:
3564:
3560:
3556:
3552:
3549:
3545:
3541:
3538:
3534:
3528:
3524:
3519:
3512:
3509:
3506:
3502:
3497:
3494:
3491:
3488:
3484:
3480:
3476:
3460:
3449:
3446:
3443:
3438:
3433:
3428:
3424:
3418:
3414:
3410:
3407:
3404:
3401:
3396:
3392:
3386:
3382:
3378:
3375:
3372:
3369:
3364:
3361:
3357:
3352:
3348:
3345:
3322:
3311:
3306:
3301:
3296:
3293:
3290:
3286:
3282:
3279:
3275:
3271:
3268:
3265:
3262:
3258:
3254:
3251:
3248:
3245:
3242:
3238:
3234:
3229:
3225:
3221:
3218:
3215:
3212:
3208:
3204:
3201:
3186:
3159:momentum space
3155:position space
3128:
3125:
3104:
3101:
3069:
3068:
3056:
3053:energy density
3031:
3028:
3024:
3020:
3017:
3014:
3008:
3005:
3000:
2997:
2975:
2972:
2971:
2970:
2943:
2938:
2934:
2930:
2925:
2921:
2917:
2901:
2877:
2865:
2844:
2841:
2835:
2832:
2827:
2824:
2818:
2814:
2810:
2807:
2804:
2801:
2795:
2792:
2787:
2784:
2778:
2773:
2769:
2762:
2759:
2754:
2751:
2745:
2740:
2736:
2729:
2726:
2721:
2718:
2696:Main article:
2693:
2690:
2672:
2669:
2666:
2662:
2658:
2655:
2644:incompressible
2621:
2620:
2605:
2597:
2572:
2569:
2566:
2562:
2558:
2555:
2552:
2549:
2546:
2540:
2537:
2532:
2529:
2515:fluid dynamics
2505:Mass flow rate
2496:
2495:Fluid dynamics
2493:
2469:
2466:
2460:
2457:
2452:
2449:
2443:
2439:
2435:
2432:
2412:
2409:
2406:
2402:
2398:
2395:
2369:
2366:
2360:
2357:
2352:
2348:
2344:
2341:
2338:
2335:
2329:
2325:
2321:
2318:
2298:
2292:
2289:
2284:
2280:
2276:
2273:
2270:
2267:
2261:
2257:
2253:
2250:
2247:
2244:
2240:
2236:
2233:
2230:
2227:
2224:
2202:
2196:
2193:
2187:
2183:
2177:
2173:
2169:
2165:
2161:
2158:
2149:, states that
2136:
2121:
2118:
2113:
2110:
2104:
2101:
2097:
2093:
2090:
2070:charge density
2033:Main article:
2030:
2027:
2015:
2012:
2008:
2004:
2001:
1998:
1992:
1989:
1984:
1981:
1921:
1920:
1880:
1872:
1840:
1820:
1794:
1791:
1787:
1783:
1780:
1777:
1771:
1768:
1763:
1760:
1747:
1728:
1725:
1694:
1693:
1674:
1669:
1666:
1662:
1658:
1655:
1589:
1581:
1567:
1553:
1536:
1532:
1527:
1523:
1512:
1503:closed surface
1470:closed surface
1444:
1441:
1437:
1433:
1430:
1426:
1420:
1416:
1412:
1406:
1403:
1398:
1395:
1382:
1366:
1365:
1352:
1333:The amount of
1331:
1318:The amount of
1311:
1308:
1306:for details.)
1296:
1295:
1278:
1274:
1269:
1265:
1239:
1235:
1232:
1228:
1222:
1218:
1214:
1211:
1208:
1200:
1192:
1182:
1181:
1180:
1179:per unit time:
1138:
1129:
1120:
1111:
1100:of a quantity
1096:
1087:
1076:
1075:
1063:
1062:
1037:
1025:
1021:
1018:
1014:
1002:
1001:
970:velocity field
968:If there is a
966:
911:be the volume
873:Main article:
870:
867:
865:
862:
858:Sankey diagram
751:
750:
748:
747:
740:
733:
725:
722:
721:
717:
716:
711:
706:
701:
696:
691:
686:
681:
676:
671:
666:
661:
656:
651:
646:
640:
639:
636:
635:
632:
631:
626:
625:
624:
623:
618:
613:
605:
604:
598:
597:
596:
595:
590:
585:
575:
570:
569:
566:
565:
560:
559:
553:
552:
550:
549:
544:
539:
534:
529:
524:
519:
513:
510:
509:
503:
502:
500:
499:
494:
489:
487:Chromatography
484:
479:
473:
470:
469:
463:
462:
460:
459:
440:
439:
438:
419:
407:
402:
390:
377:
374:
373:
365:
360:
359:
356:
355:
351:
350:
345:
340:
339:
338:
328:
323:
318:
317:
316:
311:
301:
296:
291:
286:
285:
284:
274:
268:
267:
262:
261:
258:
257:
252:
251:
250:
249:
241:
240:
236:
235:
234:
233:
228:
223:
215:
214:
208:
205:
204:
201:
200:
195:
181:
178:
173:
170:
164:
161:
158:
155:
144:
143:
137:
136:
128:
127:
42:
40:
33:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
8310:
8299:
8296:
8294:
8291:
8289:
8286:
8285:
8283:
8272:
8270:0-7167-0344-0
8266:
8262:
8261:
8255:
8251:
8245:
8241:
8236:
8232:
8230:81-7758-293-3
8226:
8222:
8217:
8213:
8207:
8203:
8202:Hydrodynamics
8198:
8197:
8193:
8184:
8182:0-7167-0344-0
8178:
8174:
8167:
8164:
8153:
8146:
8143:
8138:
8136:0-7167-0344-0
8132:
8128:
8121:
8118:
8113:
8111:0-07-145545-0
8107:
8103:
8096:
8093:
8088:
8086:0-07-145546-9
8082:
8078:
8070:
8067:
8055:
8048:
8041:
8038:
8034:
8028:
8025:
8020:
8014:
8010:
8006:
8002:
8001:
7993:
7991:
7987:
7980:
7976:
7973:
7971:
7968:
7966:
7963:
7962:
7958:
7953:
7948:
7945:
7940:
7937:
7933:
7929:
7928:
7927:
7925:
7921:
7914:
7904:
7902:
7898:
7894:
7893:baryon number
7889:
7887:
7883:
7882:
7877:
7873:
7866:
7864:
7862:
7857:
7854:
7841:
7836:
7833:
7829:
7823:
7818:
7815:
7807:
7802:
7799:
7795:
7789:
7784:
7781:
7773:
7770:
7765:
7762:
7758:
7752:
7739:
7735:
7732:
7729:However, the
7727:
7725:
7721:
7716:
7703:
7700:
7695:
7692:
7689:
7682:
7678:
7667:
7662:
7658:
7653:
7651:
7649:
7644:
7636:
7634:
7632:
7627:
7610:
7607:
7602:
7599:
7595:
7589:
7576:
7571:
7554:
7551:
7546:
7542:
7536:
7522:
7520:
7504:
7501:
7496:
7492:
7486:
7473:
7470:
7467:labeling the
7466:
7461:
7456:
7450:
7426:
7420:
7414:
7411:
7403:
7394:
7391:
7386:
7382:
7376:
7363:
7359:
7354:
7336:
7330:
7326:
7322:
7317:
7313:
7309:
7304:
7300:
7296:
7293:
7290:
7286:
7282:
7279:
7271:
7267:
7262:
7256:
7250:
7248:
7244:
7241:, especially
7240:
7234:
7226:
7221:
7219:
7213:
7211:
7193:
7189:
7185:
7180:
7176:
7169:
7161:
7157:
7153:
7148:
7143:
7139:
7130:
7126:
7122:
7116:
7113:
7108:
7105:
7099:
7094:
7090:
7086:
7080:
7077:
7072:
7069:
7063:
7058:
7054:
7050:
7044:
7041:
7036:
7033:
7021:
7003:
6999:
6995:
6990:
6986:
6979:
6975:
6968:
6965:
6960:
6957:
6949:
6945:
6941:
6938:
6935:
6930:
6926:
6922:
6919:
6915:
6908:
6905:
6901:
6894:
6891:
6886:
6880:
6877:
6872:
6869:
6857:
6841:
6838:
6833:
6830:
6822:
6818:
6814:
6811:
6808:
6803:
6799:
6795:
6792:
6789:
6784:
6780:
6770:
6768:
6743:
6739:
6735:
6730:
6726:
6719:
6713:
6710:
6705:
6702:
6694:
6691:
6686:
6684:
6676:
6673:
6668:
6665:
6655:
6652:
6648:
6640:
6636:
6632:
6627:
6623:
6616:
6611:
6608:
6602:
6595:
6589:
6586:
6583:
6580:
6574:
6571:
6566:
6563:
6557:
6551:
6545:
6541:
6537:
6535:
6527:
6524:
6520:
6512:
6508:
6504:
6499:
6495:
6488:
6483:
6480:
6474:
6467:
6461:
6458:
6452:
6449:
6446:
6443:
6437:
6433:
6429:
6427:
6422:
6419:
6415:
6409:
6406:
6401:
6398:
6382:
6364:
6353:
6342:
6329:
6325:
6321:
6316:
6310:
6305:
6299:
6296:
6290:
6287:
6283:
6277:
6275:
6271:
6268:
6251:
6247:
6239:
6236:
6233:
6232:
6231:
6212:
6208:
6204:
6199:
6195:
6188:
6180:
6176:
6167:
6162:
6149:
6145:
6141:
6135:
6127:
6118:
6113:
6109:
6105:
6099:
6091:
6080:
6076:
6072:
6069:
6066:
6060:
6052:
6034:
6028:
6025:
6019:
6017:for electrons
6016:
6012:
6006:
6004:
6000:
5997:
5995:
5977:
5973:
5965:
5962:
5959:
5958:
5957:
5938:
5934:
5930:
5925:
5921:
5914:
5906:
5902:
5893:
5888:
5875:
5871:
5867:
5861:
5853:
5844:
5839:
5835:
5831:
5825:
5817:
5806:
5802:
5798:
5795:
5789:
5781:
5766:
5764:
5760:
5759:drift current
5753:Semiconductor
5752:
5749:
5746:
5729:
5726:
5718:
5712:
5706:
5698:
5671:
5665:
5662:
5656:
5648:
5612:
5607:
5598:
5589:
5583:
5575:
5571:
5565:
5555:
5546:
5538:
5531:
5528:
5525:
5516:
5513:
5511:
5502:
5497:
5491:
5481:
5472:
5465:
5461:
5452:
5443:
5437:
5428:
5421:
5418:
5415:
5406:
5404:
5395:
5390:
5385:
5379:
5367:
5360:
5356:
5345:
5339:
5330:
5323:
5320:
5317:
5307:
5303:
5297:
5295:
5285:
5269:
5263:
5254:
5251:
5246:
5228:
5219:
5209:
5201:
5196:
5186:
5174:
5167:
5164:
5161:
5152:
5150:
5140:
5130:
5119:
5116:
5106:
5089:
5085:
5080:
5072:
5061:
5058:
5051:
5041:
5024:
5020:
5015:
5012:
5010:
5001:
4992:
4984:
4981:
4976:
4966:
4955:
4952:
4942:
4931:
4927:
4917:
4913:
4908:
4904:
4895:
4887:
4884:
4876:
4865:
4862:
4855:
4845:
4834:
4830:
4820:
4816:
4811:
4809:
4800:
4794:
4783:
4780:
4775:
4765:
4754:
4751:
4741:
4730:
4726:
4716:
4712:
4707:
4703:
4694:
4686:
4683:
4675:
4664:
4661:
4654:
4644:
4633:
4629:
4619:
4615:
4610:
4608:
4600:
4592:
4573:
4567:
4550:
4546:
4540:
4529:
4526:
4521:
4511:
4500:
4497:
4487:
4476:
4472:
4462:
4458:
4453:
4450:
4448:
4440:
4430:
4409:
4405:
4396:
4388:
4385:
4377:
4366:
4363:
4356:
4346:
4335:
4331:
4321:
4317:
4312:
4310:
4302:
4283:
4261:
4235:
4213:
4200:
4194:
4184:
4167:
4161:
4142:
4134:
4130:
4121:
4112:
4105:
4093:
4087:
4077:
4053:
4047:
4039:
4024:
4018:
4013:
4009:
4004:
3983:
3977:
3967:
3950:
3947:
3944:
3942:
3935:
3927:
3924:
3919:
3909:
3898:
3895:
3889:
3879:
3872:
3866:
3846:
3843:
3841:
3833:
3830:
3822:
3811:
3808:
3802:
3792:
3779:
3775:
3770:
3766:
3757:
3755:
3751:
3747:
3743:
3739:
3735:
3732:flows like a
3726:
3721:
3716:
3703:
3700:
3694:
3684:
3660:
3652:
3641:
3638:
3632:
3624:
3615:
3607:
3581:
3577:
3572:
3566:
3554:
3547:
3543:
3532:
3526:
3517:
3510:
3507:
3504:
3495:
3489:
3486:
3465:
3461:
3447:
3444:
3441:
3436:
3416:
3412:
3408:
3405:
3402:
3394:
3384:
3380:
3376:
3370:
3362:
3359:
3350:
3346:
3343:
3327:
3323:
3309:
3304:
3291:
3288:
3269:
3263:
3260:
3243:
3240:
3227:
3219:
3213:
3210:
3199:
3191:
3187:
3182:
3178:
3171:
3165:
3160:
3157:(rather than
3156:
3152:
3149:for a single
3145:
3141:
3140:
3139:
3137:
3133:
3126:
3124:
3122:
3118:
3114:
3110:
3102:
3100:
3098:
3097:joule heating
3094:
3090:
3086:
3082:
3081:heat equation
3078:
3077:Fourier's law
3074:
3066:
3061:
3057:
3054:
3049:
3045:
3044:
3043:
3029:
3026:
3018:
3012:
3006:
2998:
2984:
2980:
2973:
2963:
2959:
2936:
2932:
2928:
2923:
2919:
2906:
2902:
2893:
2889:
2882:
2878:
2874:
2870:
2866:
2862:
2858:
2857:
2856:
2842:
2839:
2833:
2825:
2816:
2808:
2805:
2799:
2793:
2785:
2776:
2771:
2767:
2760:
2752:
2743:
2738:
2734:
2727:
2719:
2705:
2699:
2691:
2689:
2686:
2670:
2667:
2664:
2656:
2645:
2640:
2638:
2634:
2630:
2626:
2618:
2615:
2614:flow velocity
2610:
2606:
2602:
2598:
2595:
2590:
2586:
2585:
2584:
2570:
2567:
2556:
2550:
2544:
2538:
2530:
2516:
2510:
2506:
2502:
2494:
2492:
2490:
2485:
2480:
2467:
2464:
2458:
2450:
2441:
2433:
2410:
2407:
2404:
2396:
2385:
2380:
2367:
2364:
2358:
2342:
2327:
2319:
2296:
2290:
2274:
2259:
2251:
2245:
2234:
2225:
2213:
2200:
2194:
2175:
2167:
2159:
2148:
2144:
2135:
2119:
2111:
2102:
2099:
2091:
2080:
2075:
2071:
2067:
2062:
2058:
2054:
2050:
2046:
2042:
2036:
2028:
2026:
2013:
2010:
2002:
1996:
1990:
1982:
1966:
1961:
1957:
1952:
1946:
1943:
1938:
1934:
1930:
1926:
1916:
1910:
1903:
1897:
1891:
1885:
1881:
1877:
1873:
1869:
1864:
1860:
1856:
1851:
1845:
1841:
1837:
1831:
1825:
1821:
1818:
1811:
1810:
1809:
1805:
1792:
1789:
1781:
1775:
1769:
1761:
1746:
1744:
1738:
1734:
1726:
1724:
1712:
1706:
1700:
1667:
1664:
1660:
1656:
1653:
1640:
1635:
1630:
1620:
1615:
1610:
1604:
1598:
1590:
1586:
1582:
1578:
1572:
1568:
1564:
1558:
1554:
1551:
1530:
1525:
1521:
1513:
1509:
1504:
1499:
1495:
1494:
1493:
1487:
1482:
1476:
1471:
1466:
1460:
1455:
1439:
1431:
1428:
1418:
1414:
1410:
1404:
1401:
1396:
1393:
1381:
1378:
1372:
1362:
1357:
1353:
1351:is destroyed;
1349:
1343:
1337:
1332:
1328:
1322:
1317:
1316:
1315:
1310:Integral form
1309:
1307:
1305:
1301:
1293:
1272:
1267:
1263:
1254:
1253:
1251:
1233:
1230:
1220:
1216:
1212:
1206:
1198:
1177:
1171:
1165:
1161:of flux over
1160:
1155:
1150:
1149:
1145:
1137:
1128:
1119:
1110:
1104:
1095:
1086:
1080:
1073:
1069:
1065:
1064:
1060:
1055:
1049:
1043:
1038:
1019:
1016:
1004:
1003:
997:
993:
987:
981:
975:
971:
967:
959:
952:
947:
946:
945:
943:
939:
934:
928:
922:
919:
914:
909:
904:
900:
896:
892:
887:
882:
876:
868:
863:
861:
859:
854:
852:
848:
844:
839:
837:
833:
832:flux integral
828:
824:
822:
818:
814:
810:
806:
801:
797:
792:
790:
786:
782:
778:
774:
770:
766:
762:
758:
746:
741:
739:
734:
732:
727:
726:
724:
723:
715:
712:
710:
707:
705:
702:
700:
697:
695:
692:
690:
687:
685:
682:
680:
677:
675:
672:
670:
667:
665:
662:
660:
657:
655:
652:
650:
647:
645:
642:
641:
634:
633:
622:
619:
617:
614:
612:
609:
608:
607:
606:
603:
599:
594:
591:
589:
586:
584:
581:
580:
579:
578:
573:
568:
567:
558:
554:
548:
545:
543:
540:
538:
535:
533:
530:
528:
527:Charles's law
525:
523:
520:
518:
515:
514:
512:
511:
508:
504:
498:
495:
493:
490:
488:
485:
483:
480:
478:
475:
474:
472:
471:
468:
464:
458:
455:
451:
448:
444:
441:
436:
435:non-Newtonian
433:
429:
425:
424:
423:
420:
418:
415:
411:
408:
406:
403:
401:
398:
394:
391:
389:
386:
382:
379:
378:
376:
375:
372:
368:
363:
358:
357:
349:
346:
344:
341:
337:
334:
333:
332:
329:
327:
324:
322:
321:Compatibility
319:
315:
312:
310:
309:Finite strain
307:
306:
305:
302:
300:
297:
295:
292:
290:
287:
283:
280:
279:
278:
275:
273:
270:
269:
265:
260:
259:
248:
245:
244:
243:
242:
237:
232:
229:
227:
224:
222:
219:
218:
217:
216:
213:Conservations
211:
203:
202:
198:
179:
176:
171:
168:
162:
159:
156:
153:
146:
145:
142:
138:
134:
133:
124:
121:
113:
110:December 2023
102:
99:
95:
92:
88:
85:
81:
78:
74:
71: â
70:
66:
65:Find sources:
59:
55:
49:
48:
43:This article
41:
37:
32:
31:
19:
8258:
8239:
8220:
8201:
8172:
8166:
8155:. Retrieved
8145:
8126:
8120:
8101:
8095:
8076:
8069:
8057:. Retrieved
8053:
8040:
8033:Aerodynamics
8032:
8027:
7999:
7917:
7890:
7881:color charge
7879:
7870:
7860:
7858:
7855:
7737:
7730:
7728:
7717:
7665:
7661:tensor field
7654:
7647:
7640:
7625:
7569:
7523:
7459:
7448:
7352:
7260:
7254:
7251:
7236:
7217:
7022:
6858:
6771:
6766:
6383:
6327:
6326:, along the
6323:
6319:
6317:
6314:
6300:
6291:
6278:
6269:
6234:
6038:
6029:
6020:
6007:
5998:
5960:
5767:
5756:
5747:
5267:
5261:
5247:
4571:
4568:
4259:
4233:
4214:
4022:
4016:
4002:
3777:
3773:
3762:
3754:vector field
3745:
3742:vector field
3737:
3724:
3719:
3717:
3596:
3180:
3176:
3169:
3163:
3144:wavefunction
3135:
3130:
3106:
3088:
3084:
3070:
3059:
3047:
2977:
2961:
2957:
2904:
2891:
2887:
2880:
2872:
2868:
2860:
2701:
2698:Optical flow
2641:
2622:
2617:vector field
2608:
2600:
2588:
2512:
2486:
2482:
2381:
2214:
2140:
2073:
2060:
2038:
1964:
1950:
1947:
1941:
1922:
1914:
1908:
1906:) or remove
1901:
1895:
1889:
1883:
1875:
1867:
1862:
1858:
1854:
1849:
1843:
1835:
1829:
1823:
1807:
1748:
1740:
1710:
1704:
1698:
1695:
1638:
1633:
1628:
1618:
1613:
1608:
1602:
1596:
1584:
1576:
1570:
1562:
1556:
1507:
1497:
1491:
1485:
1480:
1474:
1464:
1383:
1376:
1370:
1367:
1360:
1356:no other way
1355:
1347:
1341:
1335:
1326:
1320:
1313:
1300:flux density
1299:
1297:
1183:
1175:
1169:
1163:
1153:
1144:vector areas
1135:
1126:
1117:
1108:
1102:
1093:
1084:
1053:
1047:
1041:
995:
991:
985:
979:
973:
963:(1 g/s) / cm
957:
950:
941:
938:vector field
932:
926:
923:
917:
907:
885:
880:
878:
855:
840:
829:
825:
808:
804:
793:
760:
756:
754:
602:Smart fluids
547:Graham's law
453:
446:
431:
417:Pascal's law
413:
396:
384:
239:Inequalities
116:
107:
97:
90:
83:
76:
64:
52:Please help
47:verification
44:
8260:Gravitation
8173:Gravitation
8127:Gravitation
8059:22 December
8007:. pp.
7247:4-gradients
3326:probability
3121:teleporting
3065:energy flux
2384:Gauss's law
1157:, then the
1124:. (vectors
983:at a point
621:Ferrofluids
522:Boyle's law
294:Hooke's law
272:Deformation
8282:Categories
8157:2014-04-25
7981:References
7734:divergence
7650:divergence
7455:4-gradient
7362:divergence
7231:See also:
6311:Derivation
2685:divergence
2625:divergence
2499:See also:
2053:divergence
1817:divergence
1731:See also:
1548:denotes a
1142:represent
836:divergence
803:energy is
674:Gay-Lussac
637:Scientists
537:Fick's law
517:Atmosphere
336:frictional
289:Plasticity
277:Elasticity
80:newspapers
7837:λ
7834:μ
7824:ν
7819:λ
7816:μ
7812:Γ
7808:−
7803:ν
7800:λ
7790:μ
7785:λ
7782:μ
7778:Γ
7774:−
7766:ν
7763:μ
7753:μ
7749:∂
7696:μ
7690:ν
7683:μ
7666:covariant
7648:covariant
7603:ν
7600:μ
7590:ν
7586:∂
7575:4-current
7547:μ
7537:μ
7533:∂
7497:μ
7487:μ
7483:∂
7472:dimension
7469:spacetime
7427:⋅
7424:∇
7409:∂
7404:ρ
7401:∂
7387:μ
7377:μ
7373:∂
7294:ρ
7270:4-current
7268:called a
7243:4-vectors
7186:−
7091:μ
7055:μ
6996:−
6927:μ
6800:μ
6736:−
6633:−
6587:−
6505:−
6459:−
6354:−
6288:for holes
6248:μ
6205:−
6173:∂
6159:∂
6133:∂
6125:∂
6110:μ
6106:−
6097:∂
6089:∂
6077:μ
6070:−
6058:∂
6050:∂
5974:μ
5931:−
5899:∂
5885:∂
5859:∂
5851:∂
5836:μ
5823:∂
5815:∂
5803:μ
5787:∂
5779:∂
5719:⋅
5716:∇
5704:∂
5699:ρ
5696:∂
5686:⇒
5672:⋅
5669:∇
5666:−
5654:∂
5649:ρ
5646:∂
5604:Ψ
5595:∇
5584:∗
5580:Ψ
5576:−
5566:∗
5562:Ψ
5552:∇
5543:Ψ
5522:ℏ
5517:−
5492:∗
5488:Ψ
5478:∇
5469:Ψ
5466:−
5458:Ψ
5449:∇
5438:∗
5434:Ψ
5412:ℏ
5380:∗
5376:Ψ
5372:∇
5364:Ψ
5361:−
5353:Ψ
5350:∇
5340:∗
5336:Ψ
5314:ℏ
5304:⋅
5301:∇
5286:⋅
5283:∇
5253:operators
5250:Laplacian
5225:Ψ
5216:∇
5210:∗
5206:Ψ
5202:−
5197:∗
5193:Ψ
5183:∇
5179:Ψ
5158:ℏ
5141:∗
5137:Ψ
5127:∇
5112:Ψ
5103:ℏ
5093:ℏ
5078:Ψ
5069:∇
5052:∗
5048:Ψ
5038:ℏ
5028:ℏ
5016:−
4998:Ψ
4993:∗
4989:Ψ
4982:−
4977:∗
4973:Ψ
4963:∇
4948:Ψ
4939:ℏ
4921:ℏ
4901:Ψ
4896:∗
4892:Ψ
4882:Ψ
4873:∇
4856:∗
4852:Ψ
4842:ℏ
4835:−
4824:ℏ
4795:∗
4791:Ψ
4787:Ψ
4776:∗
4772:Ψ
4762:∇
4747:Ψ
4738:ℏ
4731:−
4720:ℏ
4708:−
4700:Ψ
4695:∗
4691:Ψ
4681:Ψ
4672:∇
4655:∗
4651:Ψ
4641:ℏ
4634:−
4623:ℏ
4598:∂
4593:ρ
4590:∂
4541:∗
4537:Ψ
4533:Ψ
4522:∗
4518:Ψ
4508:∇
4493:Ψ
4484:ℏ
4477:−
4466:ℏ
4454:−
4438:∂
4431:∗
4427:Ψ
4423:∂
4417:Ψ
4402:Ψ
4397:∗
4393:Ψ
4383:Ψ
4374:∇
4357:∗
4353:Ψ
4343:ℏ
4336:−
4325:ℏ
4300:∂
4295:Ψ
4292:∂
4284:∗
4280:Ψ
4192:∂
4185:∗
4181:Ψ
4177:∂
4171:Ψ
4159:∂
4154:Ψ
4151:∂
4143:∗
4139:Ψ
4127:Ψ
4122:∗
4118:Ψ
4103:∂
4099:∂
4085:∂
4068:Ψ
4060:∂
4045:∂
4040:ρ
4037:∂
3975:∂
3968:∗
3964:Ψ
3960:∂
3954:ℏ
3948:−
3936:∗
3932:Ψ
3920:∗
3916:Ψ
3906:∇
3886:ℏ
3880:−
3864:∂
3859:Ψ
3856:∂
3850:ℏ
3837:Ψ
3828:Ψ
3819:∇
3799:ℏ
3793:−
3728:and time
3692:∂
3675:Ψ
3667:∂
3653:⋅
3650:∇
3646:⇌
3630:∂
3625:ρ
3622:∂
3608:⋅
3605:∇
3567:∗
3563:Ψ
3559:∇
3551:Ψ
3548:−
3540:Ψ
3537:∇
3527:∗
3523:Ψ
3501:ℏ
3427:Ψ
3413:∫
3400:Ψ
3395:∗
3391:Ψ
3381:∫
3360:∈
3278:Ψ
3250:Ψ
3228:∗
3224:Ψ
3200:ρ
3167:and time
3019:⋅
3016:∇
3004:∂
2996:∂
2966:and time
2896:and time
2831:∂
2823:∂
2809:⋅
2803:∇
2791:∂
2783:∂
2758:∂
2750:∂
2725:∂
2717:∂
2657:⋅
2654:∇
2592:is fluid
2557:ρ
2551:⋅
2548:∇
2536:∂
2531:ρ
2528:∂
2501:Mass flux
2456:∂
2451:ρ
2448:∂
2434:⋅
2431:∇
2408:ρ
2397:⋅
2394:∇
2356:∂
2343:⋅
2340:∇
2334:∂
2320:⋅
2317:∇
2288:∂
2275:⋅
2272:∇
2266:∂
2252:⋅
2249:∇
2235:×
2232:∇
2226:⋅
2223:∇
2192:∂
2182:∂
2160:×
2157:∇
2117:∂
2112:ρ
2109:∂
2103:−
2092:⋅
2089:∇
2003:⋅
2000:∇
1988:∂
1983:ρ
1980:∂
1793:σ
1782:⋅
1779:∇
1767:∂
1762:ρ
1759:∂
1522:∮
1443:Σ
1429:⋅
1415:∮
1264:∬
1255:in which
1231:⋅
1217:∬
1059:discharge
1020:ρ
714:Truesdell
644:Bernoulli
593:Rheometer
588:Rheometry
428:Newtonian
422:Viscosity
172:φ
160:−
8005:Springer
7959:See also
7861:integral
7731:ordinary
7722:take in
7360:. The 4-
7266:4-vector
7233:4-vector
7214:Solution
3767:and its
3752:in this
3151:particle
3093:friction
3051:, local
2864:is time,
2604:is time,
2079:coulombs
1879:is time,
1861:, where
1721:ÎŁ < 0
1717:ÎŁ > 0
1588:is time,
903:momentum
813:teleport
811:can it "
785:momentum
775:. Since
765:equation
572:Rheology
477:Adhesion
457:Pressure
443:Buoyancy
388:Dynamics
226:Momentum
7901:isospin
7629:is the
7453:is the
7356:is the
6284:is the
6230:where:
6013:is the
5956:where:
4264:
4249:
4238:
4223:
4006:is the
2612:is the
2594:density
2141:One of
2066:amperes
2055:of the
1912:(i.e.,
1899:(i.e.,
1741:By the
1468:is any
1070:is the
913:density
805:locally
659:Charles
467:Liquids
381:Statics
326:Bending
94:scholar
8267:
8246:
8227:
8208:
8179:
8133:
8108:
8083:
8015:
7876:gluons
7872:Quarks
7623:where
7567:where
7463:is an
7444:where
7350:where
4010:. The
4000:where
3720:chance
3175:Ψ = Ψ(
3085:energy
3042:where
2855:where
2631:. The
2583:where
2507:, and
1960:energy
1917:< 0
1904:> 0
1853:(i.e.
1808:where
1614:source
1492:where
895:energy
849:, and
781:energy
763:is an
709:Stokes
704:Pascal
694:Navier
689:Newton
679:Graham
654:Cauchy
557:Plasma
452:
450:Mixing
445:
430:
412:
395:
383:
371:Fluids
304:Strain
299:Stress
282:linear
231:Energy
96:
89:
82:
75:
67:
8050:(PDF)
8009:10â13
7878:have
7465:index
6765:Here
3748:flow
3734:fluid
2983:below
1954:is a
1290:is a
936:is a
684:Hooke
664:Euler
649:Boyle
507:Gases
101:JSTOR
87:books
8265:ISBN
8244:ISBN
8225:ISBN
8206:ISBN
8177:ISBN
8131:ISBN
8106:ISBN
8081:ISBN
8061:2019
8013:ISBN
7874:and
7457:and
7245:and
5761:and
5248:The
4026:is:
3740:, a
3462:The
3324:The
3188:The
3142:The
3089:heat
2382:But
2077:(in
2064:(in
1935:and
1735:and
1634:sink
1484:can
1380:is:
1304:Flux
1133:and
1115:and
1091:and
891:mass
881:flux
875:Flux
777:mass
699:Noll
669:Fick
221:Mass
206:Laws
73:news
7888:).
7738:not
7641:In
5992:is
4252:âΨ*
4221:Ψ*
4014:of
3776:â â
3746:not
3332:at
3192:is
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1967:= 0
1962:),
1857:= Ď
1815:is
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1678:gen
1636:of
1616:of
1486:not
809:nor
759:or
56:by
8284::
8052:.
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7989:^
7899:,
7726:.
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7633:.
7521:.
7272::
6324:dx
4575::
4267:;
4247:Ψ
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3756:.
3704:0.
3179:,
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3099:.
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2871:,
2639:.
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2468:0.
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7830:T
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6603:]
6599:)
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6538:=
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6513:n
6509:R
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6492:(
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6484:e
6481:A
6475:]
6471:)
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6399:d
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6343:=
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6320:A
6303:p
6301:R
6294:p
6292:G
6281:p
6279:D
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6235:p
6218:)
6213:p
6209:R
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6192:(
6189:+
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6150:p
6146:D
6142:+
6136:x
6128:p
6119:E
6114:p
6100:x
6092:E
6081:p
6073:p
6067:=
6061:t
6053:p
6032:n
6030:R
6023:n
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5961:n
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5939:n
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5918:(
5915:+
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5868:+
5862:x
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5845:E
5840:n
5832:+
5826:x
5818:E
5807:n
5799:n
5796:=
5790:t
5782:n
5730:0
5727:=
5723:j
5713:+
5707:t
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5657:t
5613:]
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5572:)
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5514:=
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5021:1
5013:=
5002:]
4985:U
4967:2
4956:m
4953:2
4943:2
4932:+
4928:[
4918:i
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4909:+
4905:]
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4821:i
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4812:=
4801:]
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4713:1
4704:]
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4551:,
4547:]
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4332:[
4322:i
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4260:t
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4255:/
4243:Ψ
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4232:â
4229:/
4201:.
4195:t
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4135:=
4131:)
4113:(
4106:t
4094:=
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4073:|
4064:|
4054:=
4048:t
4023:t
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4003:U
3984:,
3978:t
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3945:=
3928:U
3925:+
3910:2
3899:m
3896:2
3890:2
3873:,
3867:t
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3844:=
3834:U
3831:+
3823:2
3812:m
3809:2
3803:2
3778:i
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3771:(
3730:t
3725:r
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3695:t
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3680:|
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3661:+
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3642:0
3639:=
3633:t
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3612:j
3582:.
3578:]
3573:)
3555:(
3544:)
3533:(
3518:[
3511:i
3508:m
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3496:=
3493:)
3490:t
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3479:(
3475:j
3448:.
3445:V
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3185:.
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2800:=
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2100:=
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2014:0
2011:=
2007:j
1997:+
1991:t
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