1861:
416:
2010:
22:
3734:
218:, gave at each point in space the total gravitational acceleration which would be felt by a small object at that point. This did not change the physics in any way: it did not matter if all the gravitational forces on an object were calculated individually and then added together, or if all the contributions were first added together as a gravitational field and then applied to an object.
1787:
1516:
275:âmuch like the tension in a rubber membrane. If that were the case, the observed velocity of the electromagnetic waves should depend upon the velocity of the observer with respect to the aether. Despite much effort, no experimental evidence of such an effect was ever found; the situation was resolved by the introduction of the
116:
said, "The fact that the electromagnetic field can possess momentum and energy makes it very real, and a particle makes a field, and a field acts on another particle, and the field has such familiar properties as energy content and momentum, just as particles can have." In practice, the strength of
283:
in 1905. This theory changed the way the viewpoints of moving observers were related to each other. They became related to each other in such a way that velocity of electromagnetic waves in
Maxwell's theory would be the same for all observers. By doing away with the need for a background medium,
2561:
over any finite region, we get a finite result. The infinities are not well-defined; but the finite values can be associated with the functions used as the weight functions to get the finite values, and that can be well-defined. We can define a continuous random field well enough as a
1432:
833:
2482:
Much like statistical mechanics has some overlap between quantum and classical mechanics, statistical field theory has links to both quantum and classical field theories, especially the former with which it shares many methods. One important example is
2155:
Field theory usually refers to a construction of the dynamics of a field, i.e. a specification of how a field changes with time or with respect to other independent physical variables on which the field depends. Usually this is done by writing a
214:, dealing with the force between each pair of bodies separately rapidly becomes computationally inconvenient. In the eighteenth century, a new quantity was devised to simplify the bookkeeping of all these gravitational forces. This quantity, the
3282:
1118:
2388:
which transform like vectors except for one of their components; in other words, when one rotates a vector field 360 degrees around a specific axis, the vector field turns to itself; however, spinors would turn to their negatives in the same
1510:
1732:
264:, propagated at a finite speed. Consequently, the forces on charges and currents no longer just depended on the positions and velocities of other charges and currents at the same time, but also on their positions and velocities in the past.
1298:
163:
is a vector field: specifying its value at a point in spacetime requires three numbers, the components of the gravitational field vector at that point. Moreover, within each category (scalar, vector, tensor), a field can be either a
2106:
As above with classical fields, it is possible to approach their quantum counterparts from a purely mathematical view using similar techniques as before. The equations governing the quantum fields are in fact PDEs (specifically,
666:
548:
4479:
1774:
2369:
of a crystal) specified by a tensor at each point of space. Under rotations in space, the components of the tensor transform in a more general way which depends on the number of covariant indices and contravariant
937:
1329:
1195:
2222:. Similarly for vectors, there are vector PDEs for displacement, velocity and vorticity fields in (applied mathematical) fluid dynamics, but vector calculus may now be needed in addition, being calculus for
4524:
3293:
4529:
3770:
732:
2519:; a continuous random field is a random field that has a set of functions as its index set. In particular, it is often mathematically convenient to take a continuous random field to have a
1968:
should underlie all physical phenomena, so that a classical field theory should, at least in principle, permit a recasting in quantum mechanical terms; success yields the corresponding
156:, respectively. A field has a consistent tensorial character wherever it is defined: i.e. a field cannot be a scalar field somewhere and a vector field somewhere else. For example, the
2595:
90:
700:
1219:
will create a field B, that exerts a force on nearby moving charged particles that is quantitatively different from the electric field force described above. The force exerted by
81:, i.e. a 1-dimensional (rank-1) tensor field. Field theories, mathematical descriptions of how field values change in space and time, are ubiquitous in physics. For instance, the
4148:
4517:
4153:
104:, even without referring to a test particle, a field occupies space, contains energy, and its presence precludes a classical "true vacuum". This has led physicists to consider
2359:
under rotations in space. Similarly, a dual (or co-) vector field attaches a dual vector to each point of space, and the components of each dual vector transform covariantly.
94:
2551:
1039:
1455:
1676:
117:
most fields diminishes with distance, eventually becoming undetectable. For instance the strength of many relevant classical fields, such as the gravitational field in
3763:
3877:
4512:
2453:
If there is a symmetry of the problem, not involving spacetime, under which these components transform into each other, then this set of symmetries is called an
1437:
The magnetic field is not conservative in general, and hence cannot usually be written in terms of a scalar potential. However, it can be written in terms of a
1237:
2503:
are not continuous. When dealing carefully with classical fields at finite temperature, the mathematical methods of continuous random fields are used, because
4330:
590:
3756:
1783:
was understood as a collection of two vector fields in space. Nowadays, one recognizes this as a single antisymmetric 2nd-rank tensor field in spacetime.
4375:
2173:
489:
3318:
2404:
Fields may have internal symmetries in addition to spacetime symmetries. In many situations, one needs fields which are a list of spacetime scalars: (Ï
267:
Maxwell, at first, did not adopt the modern concept of a field as a fundamental quantity that could independently exist. Instead, he supposed that the
842:
leads to the identity that gravitational field strength is identical to the acceleration experienced by a particle. This is the starting point of the
210:
that acted between any pair of massive objects. When looking at the motion of many bodies all interacting with each other, such as the planets in the
4534:
3882:
2356:
1905:
968:
fields are not only fields of force which dictate the motion of particles, but also have an independent physical reality because they carry energy.
573:
118:
4291:
339:, could be understood as the quanta of some quantum field, elevating fields to the status of the most fundamental objects in nature. That said,
2061:. As the field lines are pulled together tightly by gluons, they do not "bow" outwards as much as an electric field between electric charges.
1738:
1427:{\displaystyle \mathbf {B} (\mathbf {r} )={\frac {\mu _{0}}{4\pi }}\int {\frac {Id{\boldsymbol {\ell }}\times {\hat {\mathbf {r} }}}{r^{2}}}.}
3892:
3217:
3189:
2714:
883:
4458:
3042:
1141:
3912:
3907:
1950:
limits for diffraction. In practice though, the field theories of optics are superseded by the electromagnetic field theory of
Maxwell.
2585:
1897:
392:
Some of the simplest physical fields are vector force fields. Historically, the first time that fields were taken seriously was with
3270:
3153:
3122:
3097:
3012:
2871:
2687:
2202:(PDEs). For example, scalar PDEs might consider quantities such as amplitude, density and pressure fields for the wave equation and
221:
The development of the independent concept of a field truly began in the nineteenth century with the development of the theory of
4325:
4009:
3345:
2278:
46:
2341:) whose values are given by a single variable at each point of space. This value does not change under transformations of space.
4072:
3832:
2355:) which are specified by attaching a vector to each point of space. The components of this vector transform between themselves
3999:
2499:, are usually infinitely differentiable functions, but they are in any case almost always twice differentiable. In contrast,
828:{\displaystyle \mathbf {g} (\mathbf {r} )={\frac {\mathbf {F} (\mathbf {r} )}{m}}=-{\frac {GM}{r^{2}}}{\hat {\mathbf {r} }}.}
348:
4583:
4004:
3897:
3274:
2199:
2905:
125:
in classical electromagnetism, is inversely proportional to the square of the distance from the source (i.e. they follow
4588:
4558:
3902:
2144:
377:. Classical field theories remain useful wherever quantum properties do not arise, and can be active areas of research.
276:
203:
3944:
3436:
3430:
2957:
Gooding, David (1 January 1981). "Final Steps to the Field Theory: Faraday's Study of
Magnetic Phenomena, 1845-1850".
2108:
2026:
230:
4562:
4026:
3247:
2524:
2447:
2127:
as being solutions to their respective equations. A possible problem is that these RWEs can deal with complicated
4422:
4355:
4350:
4313:
3844:
3817:
3787:
2180:
2157:
1976:
1438:
2049:, which are polarized by the field and line up with it. This effect increases within a short distance (around 1
4279:
4274:
4183:
4143:
3859:
2466:
2092:
226:
2190:
It is possible to construct simple fields without any prior knowledge of physics using only mathematics from
315:, the quantum of the electromagnetic field. This was soon followed by the realization (following the work of
284:
this development opened the way for physicists to start thinking about fields as truly independent entities.
4468:
3622:
1980:
1883:
870:
356:
193:
2120:
1625:), there will be both an electric and a magnetic field, and both will vary in time. They are determined by
674:
4473:
4463:
4437:
4259:
4254:
4249:
4244:
4202:
4165:
4111:
3837:
3827:
3805:
3161:
2580:
2508:
2191:
1999:
1991:
1973:
1860:
1626:
1580:
1549:
980:
386:
374:
368:
340:
4390:
4370:
4308:
4237:
4016:
3994:
3969:
3869:
3205:
2500:
2496:
2476:
2247:
2161:
1780:
976:
843:
839:
420:
268:
261:
105:
2112:
2076:, the Einsteinian field theory of gravity, has yet to be successfully quantized. However an extension,
415:
4548:
4442:
4427:
4342:
4269:
4232:
4227:
4217:
3977:
3931:
3338:
2749:
2530:
We can think about a continuous random field, in a (very) rough way, as an ordinary function that is
2504:
2381:
2321:
2293:
2128:
2124:
2077:
1969:
1959:
1848:
378:
308:
241:. However, it became much more natural to take the field approach and express these laws in terms of
101:
42:
1320:
1113:{\displaystyle \mathbf {E} ={\frac {1}{4\pi \epsilon _{0}}}{\frac {q}{r^{2}}}{\hat {\mathbf {r} }}.}
4417:
4286:
4197:
4084:
4055:
3779:
3549:
3484:
2600:
2305:
2227:
2165:
1505:{\displaystyle \mathbf {B} (\mathbf {r} )={\boldsymbol {\nabla }}\times \mathbf {A} (\mathbf {r} )}
972:
450:
424:
404:
272:
257:
215:
160:
122:
2226:(as are these three quantities, and those for vector PDEs in general). More generally problems in
1727:{\displaystyle \mathbf {E} =-{\boldsymbol {\nabla }}V-{\frac {\partial \mathbf {A} }{\partial t}}}
4554:
4207:
4193:
4175:
4158:
4138:
4121:
4031:
3854:
3221:
2939:
2765:
2533:
2431:
2289:
2211:
2073:
2003:
1947:
1936:
1893:
1865:
1654:
1124:
858:
847:
352:
173:
3145:
3139:
2721:
A 'field' is any physical quantity which takes on different values at different points in space.
956:
first realized the importance of a field as a physical quantity, during his investigations into
351:(although they set it aside because of the ongoing utility of the field concept for research in
21:
2258:. The scalars (and hence the vectors, matrices and tensors) can be real or complex as both are
4380:
4116:
4094:
4021:
3982:
3959:
3266:
3213:
3185:
3181:
3175:
3149:
3118:
3093:
3038:
3008:
2930:
Weinberg, Steven (1977). "The Search for Unity: Notes for a
History of Quantum Field Theory".
2867:
2863:
2710:
2683:
2484:
2479:. As above, it can be approached by the usual infinite number of degrees of freedom argument.
2399:
2231:
2169:
2054:
1995:
1965:
1821:
324:
288:
157:
38:
2457:. One may also make a classification of the charges of the fields under internal symmetries.
25:
Illustration of the electric field surrounding a positive (red) and a negative (blue) charge.
4298:
4089:
4077:
4060:
4038:
3987:
3939:
3800:
3537:
3426:
3396:
2757:
2734:
2554:
2419:
2418:). For example, in weather prediction these may be temperature, pressure, humidity, etc. In
2366:
2250:, which are framed as matrix-tensor PDEs, and then require matrices or tensor fields, hence
2195:
2184:
2069:
2009:
1876:
1030:
948:
238:
222:
145:
141:
54:
4404:
4360:
4187:
4131:
4126:
4104:
4065:
3887:
3812:
3737:
3612:
3461:
3456:
3446:
3331:
3258:
2516:
2348:
2317:
2259:
2255:
2251:
2096:
2053:
from the vicinity of the quarks) making the color force increase within a short distance,
2034:
1932:
1869:
1805:
1600:
1566:
1534:
1293:{\displaystyle \mathbf {F} (\mathbf {r} )=q\mathbf {v} \times \mathbf {B} (\mathbf {r} ),}
975:, of the first unified field theory in physics with the introduction of equations for the
953:
393:
344:
280:
250:
234:
113:
86:
2753:
2704:
2064:
These three quantum field theories can all be derived as special cases of the so-called
1644:
Alternatively, one can describe the system in terms of its scalar and vector potentials
73:. A surface wind map, assigning an arrow to each point on a map that describes the wind
4048:
3002:
2605:
2590:
2520:
2385:
2352:
2285:
2203:
2176:. The resulting field theories are referred to as classical or quantum field theories.
2088:
2065:
1928:
1920:
1798:
1791:
1527:
1520:
1312:
1206:
1010:
992:
965:
961:
400:
396:
382:
328:
316:
246:
242:
177:
126:
82:
2143:), but these in theory can still be subjected to analytical methods given appropriate
661:{\displaystyle \mathbf {F} (\mathbf {r} )=-{\frac {GMm}{r^{2}}}{\hat {\mathbf {r} }},}
4577:
4365:
4320:
4303:
3949:
3822:
3748:
3421:
2856:
2807:
2243:
2223:
2219:
2214:. Outside of physics proper (e.g., radiometry and computer graphics), there are even
2207:
2100:
2033:
Color charge has "ternary neutral states" as well as binary neutrality (analogous to
1943:
320:
304:
296:
2769:
456:
which describes its influence on other bodies with mass. The gravitational field of
4385:
4264:
4043:
3954:
3849:
3645:
3631:
2654:
2626:
2512:
2439:
2423:
2377:
2373:
2362:
2344:
2334:
2270:
2215:
2140:
2045:
In quantum chromodynamics, the color field lines are coupled at short distances by
2014:
1670:, and from there the electric and magnetic fields are determined via the relations
543:{\displaystyle \mathbf {g} (\mathbf {r} )={\frac {\mathbf {F} (\mathbf {r} )}{m}}.}
211:
199:
137:
133:
78:
70:
2653:) known as the gauge. The retarded potential formalism requires one to choose the
2838:
838:
The experimental observation that inertial mass and gravitational mass are equal
4496:
4412:
3604:
3479:
2888:
2821:
2567:
2443:
2338:
2263:
2116:
998:
703:
435:
66:
62:
1998:) than any other theory. The two other fundamental quantum field theories are
233:
could manage with Newton-style laws that expressed the forces between pairs of
4432:
3696:
3577:
3254:
2761:
2563:
2523:
of functions as its index set, in which case the continuous random field is a
2274:
2050:
1984:
438:, which describes the gravitational force as a mutual interaction between two
292:
1983:. Quantum electrodynamics is arguably the most successful scientific theory;
3795:
3655:
3617:
3572:
3562:
3502:
2327:
2183:
in terms of the field components; the dynamics can be obtained by using the
1924:
1901:
957:
473:
58:
1786:
3706:
3650:
3582:
3517:
3403:
3391:
3363:
2558:
2472:
2304:
A convenient way of classifying a field (classical or quantum) is by the
1825:
866:
332:
109:
74:
2943:
2326:
Fields are often classified by their behaviour under transformations of
1896:, is another example of a field theory. Here the principal field is the
1769:{\displaystyle \mathbf {B} ={\boldsymbol {\nabla }}\times \mathbf {A} .}
176:
respectively. In this theory an equivalent representation of field is a
3701:
3689:
3627:
3557:
3532:
3492:
3471:
3441:
2435:
2136:
2132:
1515:
932:{\displaystyle \mathbf {g} (\mathbf {r} )=-\nabla \Phi (\mathbf {r} ).}
30:
4480:
The
Unreasonable Effectiveness of Mathematics in the Natural Sciences
4099:
3917:
3684:
3677:
3386:
3381:
3355:
3323:
2058:
1809:
1538:
1033:
tells us that the electric field due to a single charged particle is
336:
312:
153:
149:
50:
2269:
In a general setting, classical fields are described by sections of
3028:
3026:
3024:
2095:. There are different descriptions of odd classical fields both on
1190:{\displaystyle \mathbf {E} (\mathbf {r} )=-\nabla V(\mathbf {r} ).}
3714:
3672:
3665:
3660:
3497:
3451:
3413:
3371:
3225:
2427:
2239:
2046:
2018:
2008:
1859:
1785:
1514:
414:
207:
181:
20:
3177:
Quantum
Physics of Atoms, Molecules, Solids, Nuclei and Particles
16:
Physical quantities taking values at each point in space and time
3592:
3527:
3522:
3512:
3507:
3376:
2471:
Statistical field theory attempts to extend the field-theoretic
1987:
1916:
439:
300:
4535:
European
Community on Computational Methods in Applied Sciences
3752:
3327:
2179:
The dynamics of a classical field are usually specified by the
132:
A field can be classified as a scalar field, a vector field, a
108:
to be a physical entity, making the field concept a supporting
3246:. Vol. 25 (15th ed.). 1994. p. 815 – via
2641:; rather, they are only determined up to some scalar function
2288:, the most often studied fields are those that model the four
256:
The independent nature of the field became more apparent with
4530:
International
Council for Industrial and Applied Mathematics
2308:
it possesses. Physical symmetries are usually of two types:
1824:(note in nature N and S monopoles do not exist). In motion (
140:
according to whether the represented physical quantity is a
1875:), and rotating asymmetric mass-energy distributions with
1009:
based solely on its charge. We can similarly describe the
291:
were first applied to the electromagnetic field. In 1927,
2784:
172:, depending on whether it is characterized by numbers or
1629:, a set of differential equations which directly relate
271:
expressed the deformation of some underlying mediumâthe
2858:
Geons, Black Holes, and
Quantum Foam: A Life in Physics
2084:, something seldom considered in quantum field theory.
1605:
In general, in the presence of both a charge density Ï(
4149:
Numerical methods for ordinary differential equations
2536:
1741:
1679:
1458:
1332:
1240:
1144:
1042:
886:
735:
677:
593:
492:
4525:
Société de Mathématiques
Appliquées et Industrielles
4518:
Japan Society for Industrial and Applied Mathematics
4154:
Numerical methods for partial differential equations
2808:
Lecture 1 | Quantum Entanglements, Part 1 (Stanford)
1919:
can be constructed as physical fields, due to their
1127:, and hence can be described by a scalar potential,
4505:
4489:
4451:
4403:
4341:
4216:
4174:
3968:
3930:
3868:
3786:
3591:
3548:
3470:
3412:
3362:
2430:is an example of an internal symmetry, that of the
347:seriously considered Newton's pre-field concept of
2855:
2682:. London: Weidenfeld & Nicolson. p. 138.
2545:
1768:
1726:
1504:
1426:
1292:
1189:
1112:
979:. The modern version of these equations is called
931:
827:
694:
660:
542:
3138:Peskin, Michael E.; Schroeder, Daniel V. (1995).
2273:and their dynamics is formulated in the terms of
2810:, Leonard Susskind, Stanford, Video, 2006-09-25.
2596:Lagrangian and Eulerian specification of a field
2553:almost everywhere, but such that when we take a
464:in space corresponds to the ratio between force
260:'s discovery that waves in these fields, called
3180:(2nd ed.). John Wiley & Sons. p.
2996:
2994:
2992:
2925:
2923:
2921:
2919:
971:These ideas eventually led to the creation, by
434:A classical field theory describing gravity is
4513:Society for Industrial and Applied Mathematics
2680:Q is for Quantum: Particle Physics from A to Z
1816:fields due to stationary electric charges and
565:has a negligible influence on the behavior of
3764:
3339:
3088:J.A. Wheeler; C. Misner; K.S. Thorne (1973).
2976:
2974:
2972:
2735:"The Origins of the Field Concept in Physics"
2330:. The terms used in this classification are:
1843:charge (not found in nature) would induce an
69:to each point on the map, is an example of a
8:
4331:Supersymmetric theory of stochastic dynamics
2625:This is contingent on the correct choice of
2347:(such as the magnitude and direction of the
299:to successfully explain how the decay of an
2959:Historical Studies in the Physical Sciences
2206:; temperature/concentration fields for the
253:became the first to coin the term "field".
4338:
3865:
3771:
3757:
3749:
3597:
3346:
3332:
3324:
3283:"Real talk: Everything is made of fields"
2535:
1758:
1750:
1742:
1740:
1708:
1702:
1691:
1680:
1678:
1494:
1486:
1478:
1467:
1459:
1457:
1413:
1397:
1395:
1394:
1386:
1377:
1358:
1352:
1341:
1333:
1331:
1279:
1271:
1263:
1249:
1241:
1239:
1176:
1153:
1145:
1143:
1096:
1094:
1093:
1085:
1076:
1067:
1051:
1043:
1041:
918:
895:
887:
885:
811:
809:
808:
800:
786:
766:
758:
755:
744:
736:
734:
681:
679:
678:
676:
644:
642:
641:
633:
616:
602:
594:
592:
523:
515:
512:
501:
493:
491:
722:. Therefore, the gravitational field of
3037:(4th ed.). John Wiley & Sons.
2670:
2637:are not completely determined by Ï and
2618:
2495:Classical fields as above, such as the
2131:with exotic algebraic properties (e.g.
2029:). These are "colorless" combinations.
1900:, a symmetric 2nd-rank tensor field in
1652:. A set of integral equations known as
1387:
2840:The Feynman Lectures on Physics Vol II
2706:The Feynman Lectures on Physics Vol II
1935:is set . They are also subject to the
423:, mass is the source of an attractive
85:is another rank-1 tensor field, while
2890:The Feynman Lectures on Physics Vol I
2823:The Feynman Lectures on Physics Vol I
2230:may involve for example, directional
2080:, deals with quantum field theory at
1942:For electromagnetic waves, there are
1906:Newton's law of universal gravitation
1892:Einstein's theory of gravity, called
840:to an unprecedented level of accuracy
695:{\displaystyle {\hat {\mathbf {r} }}}
574:Newton's law of universal gravitation
7:
3033:M. Mansfield; C. OâSullivan (2011).
3004:McGraw Hill Encyclopaedia of Physics
2981:Kleppner, Daniel; Kolenkow, Robert.
2906:"Physics: What We Do and Don't Know"
2904:Steven Weinberg (November 7, 2013).
1990:confirm its predictions to a higher
1779:At the end of the 19th century, the
287:In the late 1920s, the new rules of
93:at each point in spacetime, or as a
3319:Particle and Polymer Field Theories
3113:I. Ciufolini; J.A. Wheeler (1995).
2566:from a space of functions into the
2164:of the field, and treating it as a
206:simply expressed the gravitational
3780:Industrial and applied mathematics
2586:Covariant Hamiltonian field theory
2540:
2218:. All these previous examples are
2172:system with an infinite number of
1715:
1705:
1572:formed by two magnetic monopoles.
1167:
912:
909:
112:of the edifice of modern physics.
77:at that point, is an example of a
61:. A weather map, with the surface
14:
4010:Stochastic differential equations
3281:Jepsen, Kathryn (July 18, 2013).
3141:An Introduction to Quantum Fields
2041:The quark/antiquark combinations.
1856:Gravitation in general relativity
865:can be rewritten in terms of the
4326:Supersymmetric quantum mechanics
3733:
3732:
3204:Giachetta, G., Mangiarotti, L.,
2279:covariant classical field theory
2091:one deals with odd fields, e.g.
1868:, mass-energy warps space time (
1759:
1751:
1743:
1709:
1692:
1681:
1495:
1487:
1479:
1468:
1460:
1398:
1342:
1334:
1280:
1272:
1264:
1250:
1242:
1177:
1154:
1146:
1097:
1044:
919:
896:
888:
853:Because the gravitational force
812:
767:
759:
745:
737:
682:
645:
603:
595:
524:
516:
502:
494:
472:exerts on a small or negligible
331:) that all particles, including
4208:Stochastic variational calculus
4000:Ordinary differential equations
3212:. Singapore: World Scientific,
3210:Advanced Classical Field Theory
3174:R. Resnick; R. Eisberg (1985).
3060:Introduction to Electrodynamics
2854:John Archibald Wheeler (1998).
2426:symmetry of the interaction of
2111:(RWEs)). Thus one can speak of
100:In the modern framework of the
4005:Partial differential equations
3878:Arbitrary-precision arithmetic
3244:Principles of Physical Science
3007:(2nd ed.). Mc Graw Hill.
2475:toward many-body systems and
2292:which one day may lead to the
2200:partial differential equations
1499:
1491:
1472:
1464:
1402:
1346:
1338:
1284:
1276:
1254:
1246:
1181:
1173:
1158:
1150:
1101:
923:
915:
900:
892:
816:
771:
763:
749:
741:
686:
649:
607:
599:
528:
520:
506:
498:
407:was then similarly described.
373:There are several examples of
89:can be formulated in terms of
1:
3893:Interactive geometry software
3275:Course of Theoretical Physics
3265:(3rd ed.). London: Pergamon.
706:lying along the line joining
561:ensures that the presence of
91:two interacting vector fields
3117:. Princeton Physics Series.
3077:(2nd ed.). p. 469.
3062:(3rd ed.). p. 326.
2983:An Introduction to Mechanics
277:special theory of relativity
204:law of universal gravitation
53:, that has a value for each
3945:Computational number theory
3908:Numerical-analysis software
2887:Richard P. Feynman (1970).
2837:Richard P. Feynman (1970).
2820:Richard P. Feynman (1970).
2546:{\displaystyle \pm \infty }
2384:to describe particles with
2234:(from which comes the term
2145:mathematical generalization
2139:, so may need calculus for
2109:relativistic wave equations
2027:gluon field strength tensor
1315:, which is determined from
231:Charles-Augustin de Coulomb
4607:
3263:Classical Theory of Fields
3144:. Westview Press. p.
2862:. London: Norton. p.
2709:. Addison Wesley Longman.
2464:
2397:
2315:
2262:in the abstract-algebraic/
1957:
1598:
1586:found in ordinary matter (
1204:
990:
946:
869:of a scalar function, the
861:, the gravitational field
483:and the test mass itself:
366:
191:
119:Newton's theory of gravity
4543:
4351:Algebra of physical space
3818:Automated theorem proving
3728:
3641:
3600:
3092:. W.H. Freeman & Co.
2893:. Addison Wesley Longman.
2843:. Addison Wesley Longman.
2826:. Addison Wesley Longman.
2762:10.1007/s00016-002-8357-5
1977:classical electrodynamics
1820:fields due to stationary
65:described by assigning a
4144:Numerical linear algebra
2910:New York Review of Books
2703:Richard Feynman (1970).
2491:Continuous random fields
2467:Statistical field theory
2461:Statistical field theory
1964:It is now believed that
1921:finite propagation speed
3883:Finite element analysis
3833:Constraint satisfaction
3248:EncyclopĂŠdia Britannica
3115:Gravitation and Inertia
2733:Ernan McMullin (2002).
1981:quantum electrodynamics
1658:allow one to calculate
871:gravitational potential
357:quantum electrodynamics
225:. In the early stages,
194:History of field theory
4438:Mathematical economics
4112:Multivariable calculus
3995:Differential equations
3838:Constraint programming
3828:Computational geometry
3162:precision tests of QED
3075:Electromagnetic Fields
2581:Conformal field theory
2547:
2509:nowhere differentiable
2192:multivariable calculus
2042:
2000:quantum chromodynamics
1933:isolated closed system
1889:
1852:
1770:
1728:
1613:) and current density
1591:
1581:magnetic dipole moment
1550:electric dipole moment
1506:
1428:
1294:
1191:
1123:The electric field is
1114:
933:
829:
696:
662:
544:
431:
369:Classical field theory
106:electromagnetic fields
26:
4391:Supersymmetry algebra
4376:Representation theory
4371:Renormalization group
4017:Differential geometry
3898:Optimization software
3870:Mathematical software
3035:Understanding Physics
3001:Parker, C.B. (1994).
2785:"Windy as forecasted"
2678:John Gribbin (1998).
2548:
2525:tempered distribution
2507:classical fields are
2505:thermally fluctuating
2501:generalized functions
2497:electromagnetic field
2477:statistical mechanics
2434:. Other examples are
2248:anisotropic diffusion
2012:
1863:
1789:
1781:electromagnetic field
1771:
1729:
1518:
1507:
1429:
1295:
1215:flowing along a path
1192:
1115:
999:charged test particle
977:electromagnetic field
934:
844:equivalence principle
830:
697:
663:
557:is much smaller than
545:
449:is associated with a
436:Newtonian gravitation
421:classical gravitation
418:
411:Newtonian gravitation
269:electromagnetic field
262:electromagnetic waves
24:
4584:Mathematical physics
4443:Mathematical finance
4428:Social choice theory
4343:Algebraic structures
4292:in quantum mechanics
4228:Analytical mechanics
4194:Stochastic processes
4166:Variational calculus
3978:Approximation theory
3903:Statistical software
3613:Evolutionary history
2534:
2515:are indexed sets of
2382:quantum field theory
2322:Spacetime symmetries
2312:Spacetime symmetries
2300:Symmetries of fields
2294:Unified Field Theory
2129:mathematical objects
2093:FaddeevâPopov ghosts
2078:thermal field theory
2055:confining the quarks
1970:quantum field theory
1960:Quantum field theory
1946:, and terms such as
1849:Conventional current
1739:
1677:
1579:field due to a pure
1456:
1330:
1238:
1142:
1040:
1005:experiences a force
884:
733:
675:
591:
490:
399:when describing the
389:are cases in point.
349:action at a distance
309:spontaneous emission
102:quantum field theory
95:single-rank 2-tensor
4589:Physical quantities
4418:Operations research
4287:Perturbation theory
4085:Multilinear algebra
4056:Functional analysis
3913:Numerical libraries
3845:Computational logic
3550:Natural environment
3259:Lifshitz, Evgeny M.
2754:2002PhP.....4...13M
2601:Scalar field theory
2394:Internal symmetries
2351:at each point in a
2242:word for stretch),
2238:, derived from the
2228:continuum mechanics
2212:diffusion equations
2082:finite temperatures
1948:near- and far-field
1655:retarded potentials
1627:Maxwell's equations
1223:on a nearby charge
981:Maxwell's equations
973:James Clerk Maxwell
960:. He realized that
451:gravitational field
445:Any body with mass
425:gravitational field
405:gravitational field
387:Maxwell's equations
273:luminiferous aether
258:James Clerk Maxwell
216:gravitational field
161:gravitational field
123:electrostatic field
75:speed and direction
41:, represented by a
4555:Mathematics portal
4452:Other applications
4176:Probability theory
4159:Validated numerics
4139:Numerical analysis
4032:Geometric analysis
4022:Differential forms
3855:Information theory
3073:Wangsness, Roald.
3058:Griffiths, David.
2543:
2432:strong interaction
2290:fundamental forces
2181:Lagrangian density
2174:degrees of freedom
2170:quantum mechanical
2125:Schrödinger fields
2074:General relativity
2043:
2004:electroweak theory
1996:significant digits
1937:inverse-square law
1927:when a simplified
1894:general relativity
1890:
1866:general relativity
1853:
1808:(black/white) and
1766:
1724:
1592:
1537:(black/white) and
1502:
1424:
1290:
1187:
1110:
929:
848:general relativity
825:
714:and pointing from
692:
658:
540:
432:
353:general relativity
227:André-Marie AmpÚre
27:
4571:
4570:
4405:Decision sciences
4399:
4398:
4381:Spacetime algebra
4073:Harmonic analysis
4039:Dynamical systems
3983:Clifford analysis
3960:Discrete geometry
3926:
3925:
3746:
3745:
3724:
3723:
3397:chemical elements
3290:Symmetry Magazine
3218:978-981-283-895-7
3206:Sardanashvily, G.
3191:978-0-471-87373-0
2716:978-0-201-02115-8
2485:mean field theory
2455:internal symmetry
2400:Internal symmetry
1966:quantum mechanics
1835:charge induces a
1722:
1419:
1405:
1372:
1211:A steady current
1104:
1091:
1074:
1029:. Using this and
846:, which leads to
819:
806:
778:
689:
652:
639:
553:Stipulating that
535:
325:Werner Heisenberg
289:quantum mechanics
239:electric currents
180:, for instance a
174:quantum operators
39:physical quantity
4596:
4356:Feynman integral
4339:
4299:Potential theory
4188:random variables
4078:Fourier analysis
4061:Operator algebra
3988:Clifford algebra
3940:Computer algebra
3866:
3773:
3766:
3759:
3750:
3736:
3735:
3608:
3598:
3538:tropical cyclone
3488:
3348:
3341:
3334:
3325:
3308:
3306:
3304:
3299:on March 4, 2016
3298:
3292:. Archived from
3287:
3273:. Vol. 2 of the
3251:
3229:
3202:
3196:
3195:
3171:
3165:
3159:
3135:
3129:
3128:
3110:
3104:
3103:
3085:
3079:
3078:
3070:
3064:
3063:
3055:
3049:
3048:
3044:978-0-47-0746370
3030:
3019:
3018:
2998:
2987:
2986:
2978:
2967:
2966:
2954:
2948:
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2927:
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2700:
2694:
2693:
2675:
2658:
2623:
2555:weighted average
2552:
2550:
2549:
2544:
2517:random variables
2420:particle physics
2196:potential theory
2185:action principle
2097:graded manifolds
2070:particle physics
1904:. This replaces
1877:angular momentum
1822:magnetic charges
1806:electric charges
1775:
1773:
1772:
1767:
1762:
1754:
1746:
1733:
1731:
1730:
1725:
1723:
1721:
1713:
1712:
1703:
1695:
1684:
1590:from monopoles).
1548:field due to an
1535:electric charges
1511:
1509:
1508:
1503:
1498:
1490:
1482:
1471:
1463:
1439:vector potential
1433:
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1028:
949:Electromagnetism
943:Electromagnetism
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930:
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891:
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598:
549:
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531:
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519:
513:
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497:
375:classical fields
363:Classical fields
235:electric charges
223:electromagnetism
4606:
4605:
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4595:
4594:
4593:
4574:
4573:
4572:
4567:
4539:
4501:
4485:
4447:
4395:
4361:Poisson algebra
4337:
4219:
4212:
4170:
4066:Operator theory
3964:
3922:
3888:Tensor software
3864:
3813:Automata theory
3782:
3777:
3747:
3742:
3720:
3637:
3606:
3587:
3544:
3486:
3466:
3457:Gaia hypothesis
3447:Plate tectonics
3408:
3358:
3352:
3315:
3302:
3300:
3296:
3285:
3280:
3241:
3238:
3236:Further reading
3233:
3232:
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3199:
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2417:
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2402:
2396:
2365:, (such as the
2357:contravariantly
2324:
2318:Global symmetry
2316:Main articles:
2314:
2302:
2256:tensor calculus
2153:
2035:electric charge
1972:. For example,
1962:
1956:
1914:
1912:Waves as fields
1870:Einstein tensor
1858:
1737:
1736:
1714:
1704:
1675:
1674:
1603:
1601:Electrodynamics
1597:
1595:Electrodynamics
1567:magnetic dipole
1562:field due to a
1454:
1453:
1409:
1379:
1364:
1354:
1328:
1327:
1321:BiotâSavart law
1236:
1235:
1209:
1203:
1140:
1139:
1081:
1063:
1056:
1038:
1037:
1017:
995:
989:
954:Michael Faraday
951:
945:
882:
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731:
730:
673:
672:
629:
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589:
588:
514:
488:
487:
413:
371:
365:
345:Richard Feynman
281:Albert Einstein
251:Michael Faraday
247:magnetic fields
196:
190:
166:classical field
114:Richard Feynman
87:electrodynamics
17:
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4051:
4049:Control theory
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3322:
3321:
3314:
3313:External links
3311:
3310:
3309:
3278:
3255:Landau, Lev D.
3252:
3250:(Macropaedia).
3237:
3234:
3231:
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3098:
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2800:
2775:
2742:Phys. Perspect
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2606:Velocity field
2603:
2598:
2593:
2591:Field strength
2588:
2583:
2577:
2575:
2572:
2542:
2539:
2521:Schwartz space
2492:
2489:
2465:Main article:
2462:
2459:
2446:and any other
2413:
2409:
2405:
2398:Main article:
2395:
2392:
2391:
2390:
2371:
2360:
2353:magnetic field
2342:
2313:
2310:
2301:
2298:
2286:modern physics
2264:ring-theoretic
2204:fluid dynamics
2152:
2149:
2101:supermanifolds
2066:standard model
2013:Fields due to
1958:Main article:
1955:
1954:Quantum fields
1952:
1944:optical fields
1929:physical model
1913:
1910:
1857:
1854:
1839:field while a
1810:magnetic poles
1777:
1776:
1765:
1761:
1757:
1753:
1749:
1745:
1734:
1720:
1717:
1711:
1707:
1701:
1698:
1694:
1690:
1687:
1683:
1599:Main article:
1596:
1593:
1539:magnetic poles
1513:
1512:
1501:
1497:
1493:
1489:
1485:
1481:
1477:
1474:
1470:
1466:
1462:
1435:
1434:
1423:
1416:
1412:
1404:
1400:
1393:
1389:
1385:
1382:
1376:
1370:
1367:
1361:
1357:
1351:
1348:
1344:
1340:
1336:
1313:magnetic field
1301:
1300:
1289:
1286:
1282:
1278:
1274:
1270:
1266:
1262:
1259:
1256:
1252:
1248:
1244:
1227:with velocity
1207:Magnetostatics
1205:Main article:
1202:
1201:Magnetostatics
1199:
1198:
1197:
1186:
1183:
1179:
1175:
1172:
1169:
1166:
1163:
1160:
1156:
1152:
1148:
1121:
1120:
1109:
1103:
1099:
1088:
1084:
1080:
1070:
1066:
1062:
1059:
1055:
1050:
1046:
1011:electric field
993:Electrostatics
991:Main article:
988:
987:Electrostatics
985:
947:Main article:
944:
941:
940:
939:
928:
925:
921:
917:
914:
911:
908:
905:
902:
898:
894:
890:
836:
835:
824:
818:
814:
803:
799:
794:
791:
785:
782:
777:
773:
769:
765:
761:
754:
751:
747:
743:
739:
688:
684:
669:
668:
657:
651:
647:
636:
632:
627:
624:
621:
615:
612:
609:
605:
601:
597:
584:) is given by
551:
550:
539:
534:
530:
526:
522:
518:
511:
508:
504:
500:
496:
412:
409:
401:electric field
397:lines of force
383:fluid dynamics
381:of materials,
367:Main article:
364:
361:
329:Wolfgang Pauli
317:Pascual Jordan
297:quantum fields
192:Main article:
189:
186:
178:field particle
83:electric field
59:space and time
15:
13:
10:
9:
6:
4:
3:
2:
4602:
4601:
4590:
4587:
4585:
4582:
4581:
4579:
4564:
4560:
4556:
4553:
4551:
4550:
4546:
4545:
4542:
4536:
4533:
4531:
4528:
4526:
4523:
4519:
4516:
4515:
4514:
4511:
4510:
4508:
4506:Organizations
4504:
4498:
4495:
4494:
4492:
4488:
4481:
4477:
4475:
4472:
4470:
4467:
4465:
4462:
4460:
4457:
4456:
4454:
4450:
4444:
4441:
4439:
4436:
4434:
4431:
4429:
4426:
4424:
4421:
4419:
4416:
4414:
4411:
4410:
4408:
4406:
4402:
4392:
4389:
4387:
4384:
4382:
4379:
4377:
4374:
4372:
4369:
4367:
4366:Quantum group
4364:
4362:
4359:
4357:
4354:
4352:
4349:
4348:
4346:
4344:
4340:
4332:
4329:
4327:
4324:
4323:
4322:
4321:Supersymmetry
4319:
4315:
4312:
4310:
4307:
4306:
4305:
4304:String theory
4302:
4300:
4297:
4293:
4290:
4289:
4288:
4285:
4281:
4278:
4276:
4273:
4271:
4268:
4266:
4263:
4261:
4258:
4256:
4253:
4251:
4248:
4247:
4246:
4243:
4239:
4236:
4234:
4231:
4230:
4229:
4226:
4225:
4223:
4221:
4215:
4209:
4206:
4204:
4203:Path integral
4201:
4199:
4195:
4192:
4189:
4185:
4184:Distributions
4182:
4181:
4179:
4177:
4173:
4167:
4164:
4160:
4157:
4155:
4152:
4150:
4147:
4145:
4142:
4141:
4140:
4137:
4133:
4130:
4128:
4125:
4123:
4120:
4118:
4115:
4114:
4113:
4110:
4106:
4103:
4101:
4098:
4096:
4093:
4091:
4088:
4087:
4086:
4083:
4079:
4076:
4075:
4074:
4071:
4067:
4064:
4062:
4059:
4058:
4057:
4054:
4050:
4047:
4045:
4042:
4041:
4040:
4037:
4033:
4030:
4028:
4025:
4023:
4020:
4019:
4018:
4015:
4011:
4008:
4006:
4003:
4001:
3998:
3997:
3996:
3993:
3989:
3986:
3985:
3984:
3981:
3979:
3976:
3975:
3973:
3971:
3967:
3961:
3958:
3956:
3953:
3951:
3950:Combinatorics
3948:
3946:
3943:
3941:
3938:
3937:
3935:
3933:
3929:
3919:
3916:
3914:
3911:
3909:
3906:
3904:
3901:
3899:
3896:
3894:
3891:
3889:
3886:
3884:
3881:
3879:
3876:
3875:
3873:
3871:
3867:
3861:
3858:
3856:
3853:
3851:
3848:
3846:
3843:
3839:
3836:
3835:
3834:
3831:
3829:
3826:
3824:
3823:Coding theory
3821:
3819:
3816:
3814:
3811:
3807:
3804:
3802:
3799:
3798:
3797:
3794:
3793:
3791:
3789:
3788:Computational
3785:
3781:
3774:
3769:
3767:
3762:
3760:
3755:
3754:
3751:
3739:
3731:
3730:
3727:
3717:
3716:
3712:
3708:
3705:
3703:
3700:
3699:
3698:
3695:
3691:
3688:
3686:
3683:
3679:
3676:
3675:
3674:
3671:
3667:
3664:
3663:
3662:
3659:
3658:
3657:
3654:
3652:
3649:
3647:
3644:
3643:
3640:
3633:
3629:
3626:
3624:
3621:
3619:
3616:
3614:
3611:
3609:
3607:(abiogenesis)
3603:
3602:
3599:
3596:
3594:
3590:
3584:
3581:
3579:
3576:
3574:
3571:
3569:
3566:
3564:
3561:
3559:
3556:
3555:
3553:
3551:
3547:
3539:
3536:
3534:
3531:
3530:
3529:
3526:
3524:
3521:
3519:
3516:
3514:
3511:
3509:
3506:
3504:
3501:
3499:
3496:
3494:
3491:
3489:
3483:
3481:
3478:
3477:
3475:
3473:
3469:
3463:
3460:
3458:
3455:
3453:
3450:
3448:
3445:
3443:
3440:
3438:
3435:
3432:
3428:
3425:
3423:
3422:Earth science
3420:
3419:
3417:
3415:
3411:
3405:
3402:
3398:
3395:
3393:
3390:
3389:
3388:
3385:
3383:
3380:
3378:
3375:
3373:
3370:
3369:
3367:
3365:
3361:
3357:
3349:
3344:
3342:
3337:
3335:
3330:
3329:
3326:
3320:
3317:
3316:
3312:
3295:
3291:
3284:
3279:
3276:
3272:
3271:0-08-016019-0
3268:
3264:
3260:
3256:
3253:
3249:
3245:
3240:
3239:
3235:
3227:
3223:
3219:
3215:
3211:
3207:
3201:
3198:
3193:
3187:
3183:
3179:
3178:
3170:
3167:
3163:
3157:
3155:0-201-50397-2
3151:
3147:
3143:
3142:
3134:
3131:
3126:
3124:0-691-03323-4
3120:
3116:
3109:
3106:
3101:
3099:0-7167-0344-0
3095:
3091:
3084:
3081:
3076:
3069:
3066:
3061:
3054:
3051:
3046:
3040:
3036:
3029:
3027:
3025:
3021:
3016:
3014:0-07-051400-3
3010:
3006:
3005:
2997:
2995:
2993:
2989:
2985:. p. 85.
2984:
2977:
2975:
2973:
2969:
2965:(2): 231â275.
2964:
2960:
2953:
2950:
2945:
2941:
2937:
2933:
2926:
2924:
2922:
2920:
2916:
2911:
2907:
2900:
2897:
2892:
2891:
2883:
2880:
2875:
2873:9780393046427
2869:
2865:
2860:
2859:
2850:
2847:
2842:
2841:
2833:
2830:
2825:
2824:
2816:
2813:
2809:
2804:
2801:
2790:
2786:
2783:SE, Windyty.
2779:
2776:
2771:
2767:
2763:
2759:
2755:
2751:
2747:
2743:
2736:
2729:
2726:
2722:
2718:
2712:
2708:
2707:
2699:
2696:
2691:
2689:0-297-81752-3
2685:
2681:
2674:
2671:
2664:
2656:
2652:
2648:
2644:
2640:
2636:
2632:
2628:
2622:
2619:
2612:
2607:
2604:
2602:
2599:
2597:
2594:
2592:
2589:
2587:
2584:
2582:
2579:
2578:
2573:
2571:
2569:
2565:
2560:
2556:
2537:
2528:
2526:
2522:
2518:
2514:
2513:Random fields
2510:
2506:
2502:
2498:
2490:
2488:
2486:
2480:
2478:
2474:
2468:
2460:
2458:
2456:
2451:
2449:
2445:
2441:
2437:
2433:
2429:
2425:
2421:
2416:
2401:
2393:
2387:
2383:
2379:
2376:(such as the
2375:
2374:spinor fields
2372:
2368:
2367:stress tensor
2364:
2363:tensor fields
2361:
2358:
2354:
2350:
2346:
2345:vector fields
2343:
2340:
2336:
2335:scalar fields
2333:
2332:
2331:
2329:
2323:
2319:
2311:
2309:
2307:
2299:
2297:
2295:
2291:
2287:
2282:
2280:
2276:
2275:jet manifolds
2272:
2271:fiber bundles
2267:
2265:
2261:
2257:
2253:
2249:
2245:
2244:complex fluid
2241:
2237:
2233:
2229:
2225:
2224:vector fields
2221:
2220:scalar fields
2217:
2213:
2209:
2205:
2201:
2197:
2193:
2188:
2186:
2182:
2177:
2175:
2171:
2167:
2163:
2159:
2150:
2148:
2146:
2142:
2141:spinor fields
2138:
2134:
2130:
2126:
2122:
2118:
2114:
2110:
2104:
2102:
2098:
2094:
2090:
2085:
2083:
2079:
2075:
2071:
2067:
2062:
2060:
2056:
2052:
2048:
2040:
2036:
2032:
2028:
2024:
2020:
2016:
2015:color charges
2011:
2007:
2005:
2001:
1997:
1993:
1989:
1986:
1982:
1978:
1975:
1971:
1967:
1961:
1953:
1951:
1949:
1945:
1940:
1938:
1934:
1930:
1926:
1925:causal nature
1922:
1918:
1911:
1909:
1907:
1903:
1899:
1898:metric tensor
1895:
1888:
1885:
1881:
1878:
1874:
1871:
1867:
1862:
1855:
1850:
1846:
1842:
1838:
1834:
1830:
1827:
1823:
1819:
1815:
1811:
1807:
1803:
1801:
1796:
1794:
1788:
1784:
1782:
1763:
1755:
1747:
1735:
1718:
1699:
1696:
1688:
1685:
1673:
1672:
1671:
1669:
1665:
1661:
1657:
1656:
1651:
1647:
1642:
1640:
1636:
1632:
1628:
1624:
1620:
1616:
1612:
1608:
1602:
1594:
1589:
1585:
1582:
1578:
1575:
1574:Bottom right:
1571:
1568:
1565:
1561:
1558:
1554:
1551:
1547:
1544:
1540:
1536:
1532:
1530:
1525:
1523:
1517:
1483:
1475:
1452:
1451:
1450:
1448:
1444:
1440:
1421:
1414:
1410:
1391:
1383:
1380:
1374:
1368:
1365:
1359:
1355:
1349:
1326:
1325:
1324:
1322:
1318:
1314:
1310:
1306:
1287:
1268:
1260:
1257:
1234:
1233:
1232:
1230:
1226:
1222:
1218:
1214:
1208:
1200:
1184:
1170:
1164:
1161:
1138:
1137:
1136:
1134:
1130:
1126:
1107:
1086:
1082:
1078:
1068:
1064:
1060:
1057:
1053:
1048:
1036:
1035:
1034:
1032:
1031:Coulomb's law
1027:
1024:
1020:
1015:
1012:
1008:
1004:
1000:
994:
986:
984:
982:
978:
974:
969:
967:
963:
959:
955:
950:
942:
926:
906:
903:
880:
879:
878:
876:
872:
868:
864:
860:
856:
851:
849:
845:
841:
822:
801:
797:
792:
789:
783:
780:
775:
752:
729:
728:
727:
725:
721:
717:
713:
709:
705:
655:
634:
630:
625:
622:
619:
613:
610:
587:
586:
585:
583:
579:
575:
572:According to
570:
568:
564:
560:
556:
537:
532:
509:
486:
485:
484:
482:
478:
475:
471:
467:
463:
459:
455:
452:
448:
443:
441:
437:
429:
426:
422:
417:
410:
408:
406:
402:
398:
395:
390:
388:
384:
380:
376:
370:
362:
360:
358:
354:
350:
346:
342:
338:
334:
330:
326:
322:
321:Eugene Wigner
318:
314:
310:
306:
305:quantum state
302:
298:
294:
290:
285:
282:
278:
274:
270:
265:
263:
259:
254:
252:
248:
244:
240:
236:
232:
228:
224:
219:
217:
213:
209:
205:
201:
195:
187:
185:
183:
179:
175:
171:
170:quantum field
167:
162:
159:
155:
151:
147:
143:
139:
135:
130:
128:
124:
120:
115:
111:
107:
103:
98:
96:
92:
88:
84:
80:
76:
72:
68:
64:
60:
56:
52:
48:
44:
40:
36:
32:
23:
19:
4561: /
4557: /
4547:
4423:Optimization
4386:Superalgebra
4245:Field theory
4218:Mathematical
4196: /
4044:Chaos theory
4027:Gauge theory
3955:Graph theory
3850:Cryptography
3713:
3646:Biodiversity
3632:astrobiology
3567:
3354:Elements of
3301:. Retrieved
3294:the original
3289:
3262:
3243:
3209:
3200:
3176:
3169:
3140:
3133:
3114:
3108:
3089:
3083:
3074:
3068:
3059:
3053:
3034:
3003:
2982:
2962:
2958:
2952:
2938:(4): 17â35.
2935:
2931:
2909:
2899:
2889:
2882:
2857:
2849:
2839:
2832:
2822:
2815:
2803:
2792:. Retrieved
2788:
2778:
2748:(1): 13â39.
2745:
2741:
2728:
2720:
2705:
2698:
2679:
2673:
2655:Lorenz gauge
2650:
2646:
2642:
2638:
2634:
2630:
2621:
2568:real numbers
2529:
2494:
2481:
2470:
2454:
2452:
2440:weak isospin
2414:
2403:
2378:Dirac spinor
2325:
2303:
2283:
2268:
2235:
2216:light fields
2189:
2178:
2154:
2151:Field theory
2121:KleinâGordon
2105:
2086:
2081:
2063:
2044:
2038:
2030:
2022:
1985:experimental
1963:
1941:
1915:
1891:
1886:
1879:
1872:
1844:
1840:
1836:
1832:
1828:
1817:
1813:
1812:(red/blue).
1799:
1792:
1778:
1667:
1663:
1659:
1653:
1649:
1645:
1643:
1638:
1634:
1630:
1622:
1618:
1614:
1610:
1606:
1604:
1587:
1583:
1576:
1573:
1569:
1564:mathematical
1563:
1559:
1557:Bottom left:
1556:
1552:
1545:
1542:
1541:(red/blue).
1528:
1521:
1446:
1442:
1436:
1316:
1308:
1304:
1302:
1228:
1224:
1220:
1216:
1212:
1210:
1132:
1128:
1125:conservative
1122:
1025:
1022:
1018:
1013:
1006:
1002:
1001:with charge
996:
970:
952:
874:
862:
859:conservative
854:
852:
837:
723:
719:
715:
711:
707:
670:
581:
577:
571:
566:
562:
558:
554:
552:
480:
476:
469:
465:
461:
457:
453:
446:
444:
433:
427:
391:
372:
341:John Wheeler
286:
266:
255:
220:
212:Solar System
200:Isaac Newton
197:
169:
165:
138:tensor field
134:spinor field
131:
99:
79:vector field
71:scalar field
34:
28:
18:
4563:topics list
4497:Mathematics
4413:Game theory
4314:Topological
4280:Topological
4275:Statistical
4238:Hamiltonian
3697:Prokaryotes
3485:Atmosphere
3480:Meteorology
3160:. Also see
3090:Gravitation
2557:of all the
2444:strangeness
2380:) arise in
2339:temperature
2162:Hamiltonian
2089:BRST theory
1666:from Ï and
704:unit vector
479:located at
460:at a point
307:led to the
303:to a lower
127:Gauss's law
63:temperature
4578:Categories
4469:Psychology
4433:Statistics
4233:Lagrangian
3860:Statistics
3796:Algorithms
3578:Wilderness
3431:geological
3242:"Fields".
2794:2021-06-25
2789:Windy.com/
2665:References
2564:linear map
2559:infinities
2450:symmetry.
2306:symmetries
2232:elasticity
2158:Lagrangian
2113:YangâMills
2017:, like in
1974:quantizing
1884:GEM fields
379:Elasticity
293:Paul Dirac
249:; in 1845
4474:Sociology
4464:Chemistry
4260:Effective
4255:Conformal
4250:Classical
4122:Geometric
4095:Geometric
3656:Eukaryota
3623:Hierarchy
3618:Biosphere
3583:Wildfires
3573:Radiation
3563:Ecosystem
3503:Moonlight
3437:Structure
3392:particles
3226:0811.0331
2541:∞
2538:±
2337:(such as
2328:spacetime
2246:flows or
2166:classical
1994:(to more
1992:precision
1902:spacetime
1882:generate
1756:×
1752:∇
1716:∂
1706:∂
1700:−
1693:∇
1689:−
1637:to Ï and
1484:×
1480:∇
1403:^
1392:×
1388:ℓ
1375:∫
1369:π
1356:μ
1311:) is the
1269:×
1168:∇
1165:−
1102:^
1065:ϵ
1061:π
958:magnetism
913:Φ
910:∇
907:−
817:^
784:−
687:^
650:^
614:−
474:test mass
394:Faraday's
333:electrons
158:Newtonian
4549:Category
4198:analysis
4117:Exterior
4090:Exterior
3970:Analysis
3932:Discrete
3806:analysis
3738:Category
3707:bacteria
3690:protista
3651:Organism
3518:Sunlight
3364:Universe
3261:(1971).
2944:20024506
2932:Daedalus
2770:27691986
2574:See also
2473:paradigm
2370:indices.
2135:are not
2002:and the
1851:is used.
1841:magnetic
1833:electric
1826:velocity
1016:so that
966:magnetic
962:electric
867:gradient
243:electric
110:paradigm
4559:outline
4490:Related
4459:Biology
4309:Bosonic
4270:Quantum
4220:physics
4186: (
3918:Solvers
3715:Viruses
3702:archaea
3678:animals
3630: (
3628:Biology
3605:Origin
3558:Ecology
3533:tornado
3493:Climate
3487:(Earth)
3472:Weather
3442:Geology
3429: (
3427:History
3303:June 9,
3208:(2009)
2750:Bibcode
2448:flavour
2436:isospin
2412:, ... Ï
2266:sense.
2137:tensors
2133:spinors
2059:hadrons
2057:within
2039:Bottom:
2025:is the
1847:field.
1804:due to
1533:due to
1319:by the
337:protons
188:History
152:, or a
121:or the
97:field.
31:science
4132:Vector
4127:Tensor
4105:Vector
4100:Tensor
3801:design
3666:plants
3498:Clouds
3462:Future
3452:Oceans
3404:Change
3387:Matter
3382:Energy
3356:nature
3269:
3216:
3188:
3152:
3121:
3096:
3041:
3011:
2942:
2870:
2768:
2713:
2686:
2428:quarks
2422:, the
2260:fields
2252:matrix
2236:tensor
2047:gluons
2019:quarks
1979:gives
1931:of an
1831:), an
1802:fields
1795:fields
1531:fields
1524:fields
1303:where
671:where
440:masses
403:. The
327:, and
313:photon
202:, his
154:tensor
150:spinor
146:vector
142:scalar
67:number
51:tensor
47:vector
43:scalar
4265:Gauge
3685:fungi
3673:fauna
3661:flora
3568:Field
3523:Tides
3414:Earth
3372:Space
3297:(PDF)
3286:(PDF)
3222:arXiv
2940:JSTOR
2766:S2CID
2738:(PDF)
2627:gauge
2613:Notes
2424:color
2389:case.
2349:force
2240:Latin
2160:or a
2117:Dirac
1917:Waves
702:is a
468:that
311:of a
295:used
208:force
182:boson
168:or a
136:or a
55:point
49:, or
37:is a
35:field
3593:Life
3528:Wind
3513:Snow
3508:Rain
3377:Time
3305:2015
3267:ISBN
3257:and
3214:ISBN
3186:ISBN
3150:ISBN
3119:ISBN
3094:ISBN
3039:ISBN
3009:ISBN
2868:ISBN
2711:ISBN
2684:ISBN
2633:and
2386:spin
2320:and
2208:heat
2198:and
2123:and
2099:and
2031:Top:
1988:data
1923:and
1797:and
1790:The
1662:and
1648:and
1633:and
1543:Top:
1526:and
1519:The
964:and
710:and
385:and
355:and
343:and
335:and
301:atom
245:and
229:and
148:, a
144:, a
33:, a
3182:684
3146:198
2936:106
2864:163
2758:doi
2408:, Ï
2284:In
2281:).
2254:or
2168:or
2087:In
2068:of
2037:).
1864:In
1588:not
1449:):
1231:is
1135:):
877:):
857:is
726:is
718:to
419:In
359:).
279:by
237:or
198:To
129:).
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2991:^
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