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Field (physics)From Wikipedia, the free encyclopedia
The magnitude and direction of a two-dimensional electric field surrounding two equally charged (repelling) particles.
Brightness represents magnitude and hue represents direction.
Oppositely charged (attracting) particles.
A field is aphysical quantitythat has a value for each point inspaceandtime.[1]
For example, in a weather
forecast, the wind velocity during a day over a country is described by assigning a vector to each point in
space. Each vector represents the direction of the movement of air at that point. As the day progresses, the
directions in which the vectors point change as the directions of the wind change.
A field can be classified as ascalar field, avector field, aspinor field, or atensor fieldaccording to whether the
value of the field at each point is ascalar, avector, aspinor(e.g., a Dirac electron) or, more generally, atensor,
respectively. For example, theNewtoniangravitational fieldis 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 classical fieldor a quantum field, depending
on whether it is characterized by numbers orquantum operatorsrespectively.
A field may be thought of as extending throughout the whole of space. In practice, the strength of every known
field has been found to diminish with distance to the point of being undetectable. For instance, inNewton'stheory of gravity, the gravitational field strength is inversely proportional to the square of the distance from the
gravitating object. Therefore the Earth's gravitational field quickly becomes undetectable (oncosmicscales).
Defining the field as "numbers in space" shouldn't detract from the idea that it hasphysicalreality.It occupies
space. It contains energy. Its presence eliminates a true vacuum.[2]The field creates a "condition in
space"[3]
such that when we put a particle in it, the particle "feels" a force.
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If an electrical charge is accelerated, the effects on another charge do not appear instantaneously. The first
charge feels areactionforce, picking upmomentum, but the second charge feels nothing until the influence,
traveling at thespeed of light, reaches it and gives it the momentum. Where is the momentum before the
second charge moves? By the law ofconservation of momentumit must be somewhere. Physicists have found
it of "great utility for the analysis of forces"[3]to think of it as being in the field.
This utility leads to physicists believing thatelectromagnetic fieldsactually exist, making the field concept a
supportingparadigmof the entire edifice of modern physics. That said,John WheelerandRichard
Feynmanhave entertained Newton's pre-field concept ofaction at a distance(although they put it on the back
burner because of the ongoing utility of the field concept for research ingeneral relativityandquantum
electrodynamics).
"The fact that the electromagnetic field can possess momentum and energy makes it very real... 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".[3]
Contents
[hide]
1 History
2 Classical fields
o 2.1 Newtonian gravitation
o 2.2 Electromagnetism
2.2.1 Electrostatics
2.2.2 Magnetostatics
2.2.3 Electrodynamics
o 2.3 Gravitation in general relativity
o 2.4 Waves as fields
3 Quantum fields
4 Field theory
o 4.1 Symmetries of fields
4.1.1 Spacetime symmetries
4.1.2 Internal symmetries
o 4.2 Statistical field theory
o 4.3 Continuous random fields
o 4.4 Mathematics of fields
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ki/Field_(physics)#Gravitation_in_general_relativityhttps://en.wikipedia.org/wiki/Field_(physics)#Electrodynamicshttps://en.wikipedia.org/wiki/Field_(physics)#Magnetostaticshttps://en.wikipedia.org/wiki/Field_(physics)#Electrostaticshttps://en.wikipedia.org/wiki/Field_(physics)#Electromagnetismhttps://en.wikipedia.org/wiki/Field_(physics)#Newtonian_gravitationhttps://en.wikipedia.org/wiki/Field_(physics)#Classical_fieldshttps://en.wikipedia.org/wiki/Field_(physics)#Historyhttps://en.wikipedia.org/wiki/Field_(physics)https://en.wikipedia.org/wiki/Field_(physics)#cite_note-Feynman-3https://en.wikipedia.org/wiki/Quantum_electrodynamicshttps://en.wikipedia.org/wiki/Quantum_electrodynamicshttps://en.wikipedia.org/wiki/General_relativityhttps://en.wikipedia.org/wiki/Action_at_a_distance_(physics)https://en.wikipedia.org/wiki/Richard_Feynmanhttps://en.wikipedia.org/wiki/Richard_Feynmanhttps://en.wikipedia.org/wiki/John_Archibald_Wheelerhttps://en.wikipedia.org/wiki/Paradigmhttps://en.wikipedia.org/wiki/Electromagnetic_fieldhttps://en.wikipedia.org/wiki/Field_(physics)#cite_note-Feynman-3https://en.wikipedia.org/wiki/Conservation_of_momentumhttps://en.wikipedia.org/wiki/Speed_of_lighthttps://en.wikipedia.org/wiki/Momentumhttps://en.wikipedia.org/wiki/Reaction_(physics)7/28/2019 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5 See also
6 Notes
7 References
8 Further reading
9 External links
History [edit]
The first field to appear in physics was thegravitational field. ToIsaac Newtonhislaw of universal
gravitationsimply expressed the gravitationalforcethat acted between any pair of massive objects. In the
eighteenth century, a new entity was devised to simplify the bookkeeping of all these gravitational forces. This
entity, the gravitational field, gave at each point in space the total gravitational force on an object with unit mass
at that point. This did not change the physics in any way: it did not matter if you calculated all the gravitational
forces on an object individually and then added them together, or if you first added all the contributions together
as a gravitational field and then applied it to an object.[4]
The development of the independent concept of a field truly began in the nineteenth century with the
development of the theory ofelectromagnetism. In the early stages,Andr-Marie AmpreandCharles-
Augustin de Coulombcould manage with Newton-style laws that expressed the forces between pairs ofelectric
chargesorelectric currents. However, it became much more natural to take the field approach and express
these laws in terms ofelectricandmagnetic fields; in 1849Michael Faradaybecame the first to coin the term
"field".[4]
The independent nature of the field became more apparent withJames Clerk Maxwell's discovery thatwaves in
these fieldspropagated 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.[4]
Maxwell, at first, did not adopt the modern concept of a field as fundamental entity that could independently
exist. Instead he supposed that theelectromagnetic fieldexpressed the deformation of some underlying
mediumtheluminiferous aethermuch like the tension in a rubber membrane. A direct consequence of this
hypothesis was that the observed velocity of the electromagnetic waves should depend on 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 thetheory of special relativitybyAlbert Einsteinin 1905.
This theory changed the way the viewpoints of moving observers should be 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, this development opened the way for physicists to start thinking about
fields as truly independent entities.[4]
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chael_Faradayhttps://en.wikipedia.org/wiki/Magnetic_fieldhttps://en.wikipedia.org/wiki/Electric_fieldhttps://en.wikipedia.org/wiki/Electric_currenthttps://en.wikipedia.org/wiki/Electric_chargehttps://en.wikipedia.org/wiki/Electric_chargehttps://en.wikipedia.org/wiki/Charles-Augustin_de_Coulombhttps://en.wikipedia.org/wiki/Charles-Augustin_de_Coulombhttps://en.wikipedia.org/wiki/Andr%C3%A9-Marie_Amp%C3%A8rehttps://en.wikipedia.org/wiki/Electromagnetismhttps://en.wikipedia.org/wiki/Field_(physics)#cite_note-Weinberg1977-4https://en.wikipedia.org/wiki/Forcehttps://en.wikipedia.org/wiki/Law_of_universal_gravitationhttps://en.wikipedia.org/wiki/Law_of_universal_gravitationhttps://en.wikipedia.org/wiki/Isaac_Newtonhttps://en.wikipedia.org/wiki/Gravitational_fieldhttps://en.wikipedia.org/w/index.php?title=Field_(physics)&action=edit§ion=1https://en.wikipedia.org/wiki/Field_(physics)#External_linkshttps://en.wikipedia.org/wiki/Field_(physics)#Further_readinghttps://en.wikipedia.org/wiki/Field_(physics)#Referenceshttps://en.wikipedia.org/wiki/Field_(physics)#Noteshttps://en.wikipedia.org/wiki/Field_(physics)#See_also7/28/2019 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In the late 1920s, the new rules ofquantum mechanicswere first applied to the electromagnetic fields. In
1927,Paul Diracusedquantum fieldsto successfully explain how the decay of anatomto lowerquantum
statelead to thespontaneous emissionof aphoton, the quantum of the electromagnetic field. This was soon
followed by the realization (following the work ofPascual Jordan,Eugene Wigner,Werner Heisenberg,
andWolfgang Pauli) that all particles includingelectronsandprotonscould be understood as the quanta of
some quantum field, elevating fields to the most fundamental objects in nature.[4]
Classical fields [edit]
Main article:Classical field theory
There are several examples ofclassical fields. Classical field theories remain useful wherever quantum
properties do not arise, and can be active areas of research.Elasticityof materials,fluid
dynamicsandMaxwell's equationsare cases in point.
Some of the simplest physical fields are vector force fields. Historically, the first time that fields were taken
seriously was withFaraday'slines of forcewhen describing theelectric field. Thegravitational fieldwas then
similarly described.
Newtonian gravitation [edit]
Inclassical gravitation, mass is the source of an attractivegravitational fieldg.
A classical field theory describing gravity isNewtonian gravitation, which describes the gravitational force as a
mutual interaction between twomasses.
Any massive body Mhas agravitational fieldg which describes its influence on other massive bodies. The
gravitational field ofMat a point rin space is found by determining the force F that Mexerts on a smalltest
massm located at r, and then dividing by m:[5]
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Stipulating that m is much smaller than Mensures that the presence ofm has a negligible influence the
behavior ofM.
According toNewton's law of gravitation,F(r) is given by[5]
where is aunit vectorlying along the line joining Mand m and pointing from m to M. Therefore, the
gravitational field ofM is[5]
The experimental observation that inertial mass and gravitational mass are equal
tounprecedented levels of accuracyleads to the identification of the gravitational field strength as
identical to the acceleration experienced by a particle. This is the starting point of theequivalenceprinciple, which leads togeneral relativity.
Because the gravitational force F isconservative, the gravitational field g can be rewritten in
terms of thegradientof agravitational potential(r):
Electromagnetism [edit]
Main article:Electromagnetism
Michael Faradayfirst realized the importance of a field as a physical object, during his
investigations intomagnetism. He realized thatelectricandmagneticfields are not only fields
of force which dictate the motion of particles, but also have an independent physical reality
because they carry energy.
These ideas eventually led to the creation, byJames Clerk Maxwell, of the first unified field
theory in physics with the introduction of equations for theelectromagnetic field. The modern
version of these equations is calledMaxwell's equations.
Electrostatics [edit]
Main article:Electrostatics
Acharged test particlewith charge q experiences a force F based solely on its charge. We
can similarly describe theelectric fieldE so that F = qE. Using this andCoulomb's lawtells us
that the electric field due to a single charged particle as
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The electric field isconservative, and hence can be described by a scalar potential, V(r):
Magnetostatics [edit]
Main article:Magnetostatics
A steady current Iflowing along a path will exert a force on nearby charged
particles that is quantitatively different from the electric field force described above.
The force exerted by Ion a nearby charge q with velocity v is
where B(r) is themagnetic field, which is determined from Iby theBiot-Savart
law:
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 avector potential,A(r):
E fields due to stationary electric charges and B fields due to
stationarymagnetic charges. In motion (velocityv), an electriccharge induces
a Bfield while a magneticcharge induces an E field.Conventional currentis
used.
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Top:E field due to anelectric dipole momentd. Bottom left:B field due to
a mathematicalmagnetic dipolem formed by two magnetic
monopoles. Bottom right:B field due to a puremagnetic dipole
momentm found in ordinary matter (notfrom monopoles).
TheE fieldsandB fieldsdue toelectric charges(black/white) andmagnetic
poles(red/blue).[6][7]
Electrodynamics [edit]
Main article:Electrodynamics
In general, in the presence of both a charge density (r, t) and current
density J(r, t), there will be both an electric and a magnetic field, and
both will vary in time. They are determined byMaxwell's equations, a
set of differential equations which directly relate E and Bto andJ.[8]
Alternatively, one can describe the system in terms of its scalar and
vector potentials Vand A. A set of integral equations known
asretarded potentialsallow one to calculate VandAfrom andJ,[note
1]and from there the electric and magnetic fields are determined via
the relations[9]
At the end of the 19th century, theelectromagnetic fieldwas
understood as a collection of two vector fields in space.
Nowadays, one recognizes this as a single antisymmetric
2nd-rank tensor field in spacetime.
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Gravitation in general relativity [edit]
Ingeneral relativity, mass-energy warps space time (Einstein
tensorG),[10]and rotating asymmetric mass-energy distributions
withangular momentumJ generateGEM fieldsH[11]
Einstein's theory of gravity, calledgeneral relativity, is
another example of a field theory. Here the principal field is
themetric tensor, a symmetric 2nd-rank tensor field
inspacetime. This replacesNewton's law of universal
gravitation.
Waves as fields [edit]
Wavescan be constructed as physical fields, due to
theirfinite propagation speedandcausal naturewhen a
simplifiedphysical modelof anisolated closed systemis
set[clarification needed]
. They are also subject to theinverse-square
law.
For electromagnetic waves, there areoptical fields, and
terms such asnear- and far-fieldlimits for diffraction. In
practice, though the field theories of optics are superseded
by the electromagnetic field theory of Maxwell.
Quantum fields [edit]
Main article:Quantum field theory
Further information:Wavefunction collapse
It is now believed thatquantum mechanicsshould underlie
all physical phenomena, so that a classical field theory
should, at least in principle, permit a recasting in quantum
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i/Physical_modelhttps://en.wikipedia.org/wiki/Causalityhttps://en.wikipedia.org/wiki/Speed_of_lighthttps://en.wikipedia.org/wiki/Waveshttps://en.wikipedia.org/w/index.php?title=Field_(physics)&action=edit§ion=9https://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitationhttps://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitationhttps://en.wikipedia.org/wiki/Spacetimehttps://en.wikipedia.org/wiki/Metric_tensor_(general_relativity)https://en.wikipedia.org/wiki/General_relativityhttps://en.wikipedia.org/wiki/Field_(physics)#cite_note-12https://en.wikipedia.org/wiki/Gravitoelectromagnetismhttps://en.wikipedia.org/wiki/Angular_momentumhttps://en.wikipedia.org/wiki/Field_(physics)#cite_note-11https://en.wikipedia.org/wiki/Einstein_tensorhttps://en.wikipedia.org/wiki/Einstein_tensorhttps://en.wikipedia.org/wiki/General_relativityhttps://en.wikipedia.org/w/index.php?title=Field_(physics)&action=edit§ion=87/28/2019 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mechanical terms; success yields the
correspondingquantum field theory. For
example,quantizingclassical
electrodynamicsgivesquantum electrodynamics. Quantum
electrodynamics is arguably the most successful scientific
theory;experimentaldataconfirm its predictions to a
higherprecision(to moresignificant digits) than any other
theory.[12]The two other fundamental quantum field theories
arequantum chromodynamicsand theelectroweak theory.
Fields due tocolor charges, like inquarks(G is thegluon field
strength tensor). These are "colorless" combinations. Top: Color
charge has "ternary neutral states" as well as binary neutrality
(analogous toelectric charge). Bottom: The quark/antiquark
combinations.[6][7]
In quantum chromodynamics, the color field lines are
coupled at short distances bygluons, which are polarized by
the field and line up with it. This effect increases within a
short distance (around 1fmfrom the vicinity of the quarks)
making the color force increase within a short
distance,confining the quarkswithinhadrons. As the field
lines are pulled together tightly by gluons, they do not "bow"
outwards as much as an electric field between electric
charges.[13]
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These three quantum field theories can all be derived as
special cases of the so-calledstandard modelofparticle
physics.General relativity, the Einsteinian field theory of
gravity, has yet to be successfully quantized. However an
extension,thermal field theory, deals with quantum field
theory at finite temperatures, something seldom considered
in quantum field theory.
InBRST theoryone deals with odd fields, e.g.ghosts. There
are different descriptions of odd classical fields both
ongraded manifoldsandsupermanifolds.
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 (more precisely,relativistic
wave equations(RWEs)). Thus one can speak ofYang-
Mills,Dirac,Klein-GordonandSchroedinger fieldsas being
solutions to their respective equations. A possible problem is
that these RWEs can deal with complicatedmathematical
objectswith exotic algebraic properties (e.g.spinorsare
nottensors, so may need calculus overspinor fields), but
these in theory can still be subjected to analytical methods
given appropriatemathematical generalization.
Some theories, such as theBatalinVilkovisky formalism,
contains both fields and antifields.
Field theory [edit]
A field theory is aphysical theorythat describes how one or
more physical fields interact with matter.
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
aLagrangianor aHamiltonianof the field, and treating it as
theclassical mechanics(orquantum mechanics) of a system
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/Supermanifoldhttps://en.wikipedia.org/wiki/Graded_manifoldhttps://en.wikipedia.org/wiki/Faddeev%E2%80%93Popov_ghosthttps://en.wikipedia.org/wiki/BRST_formalismhttps://en.wikipedia.org/wiki/Thermal_field_theoryhttps://en.wikipedia.org/wiki/General_relativityhttps://en.wikipedia.org/wiki/Particle_physicshttps://en.wikipedia.org/wiki/Particle_physicshttps://en.wikipedia.org/wiki/Standard_model7/28/2019 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with an infinite number ofdegrees of freedom. The resulting
field theories are referred to as classical or quantum field
theories.
The dynamics of a classical field are usually specified by
theLagrangian densityin terms of the field components; the
dynamics can be obtained by using theaction principle.
It is possible to construct simple fields without any a priori
knowledge of physics using only mathematics fromseveral
variable calculus, potential theory and partial differential
equations. For example, scalar PDEs might consider
quantities such as amplitude, density and pressure fields for
the wave equation andfluid dynamics;
temperature/concentration fields for theheat/diffusion
equations. Outside of physics proper (e.g., radiometry and
computer graphics), there are evenlight fields. All these
previous examples arescalar fields. 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
overvector fields(as are these three quantities, and those
for vector PDEs in general). More generally problems
incontinuum mechanicsmay involve for example,
directionalelasticity(from which comes the term tensor,
derived from theLatinword for stretch),complex fluidflows
oranisotropic diffusion, which are framed as matrix-tensor
PDEs, and then require matrices or tensor fields,
hencematrixortensor calculus. It should be noted that the
scalars (and hence the vectors, matrices and tensors) can
be real or complex as both arefieldsin the abstract-
algebraic/ring-theoreticsense.
In a general setting, classical fields are described by
sections offiber bundlesand their dynamics is formulated in
the terms ofjet manifolds(covariant classical field theory).[14]
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Inmodern physics, the most often studied fields are those
that model the fourfundamental forceswhich one day may
lead to theUnified Field Theory.
Symmetries of fields [edit]
A convenient way of classifying a field (classical or quantum)
is by thesymmetriesit possesses. Physical symmetries are
usually of two types:
Spacetime symmetries [edit]
Main article:Spacetime symmetries
Fields are often classified by their behaviour under
transformations ofspacetime. The terms used in this
classification are
scalar fields(such astemperature) whose values are
given by a single variable at each point of space. This
value does not change under transformations of space.
vector fields(such as the magnitude and direction of
theforceat each point in amagnetic field) which are
specified by attaching a vector to each point of space.
The components of this vector transform between
themselves as usual under rotations in space.
tensor fields, (such as thestress tensorof a crystal)
specified by a tensor at each point of space. The
components of the tensor transform between
themselves as usual under rotations in space.
spinor fields(such as theDirac spinor) arise inquantum
field theoryto describe particles withspin.
Internal symmetries [edit]
Main article:Gauge symmetry
Fields may have internal symmetries in addition to spacetime
symmetries. For example, in many situations one needs
fields which are a list of space-time scalars: (1, 2, ... N).
For example, in weather prediction these may be
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temperature, pressure, humidity, etc. Inparticle physics,
thecolorsymmetry of the interaction ofquarksis an example
of an internal symmetry of thestrong interaction, as is
theisospinorflavoursymmetry.
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 internal
symmetry. One may also make a classification of the
charges of the fields under internal symmetries.
Statistical field theory [edit]
Main article:Statistical field theory
Statistical field theory attempts to extend the field-
theoreticparadigmtoward many body systems andstatistical
mechanics. As above, it can be approached by the usual
infinite number of degrees of freedom argument.
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 ismean field theory.
Continuous random fields [edit]
Classical fields as above, such as theelectromagnetic field,
are usually infinitely differentiable functions, but they are in
any case almost always twice differentiable. In
contrast,generalized functionsare not continuous. When
dealing carefully with classical fields at finite temperature,
the mathematical methods of continuous random fields are
used, becausethermally fluctuatingclassical fields
arenowhere differentiable.Random fieldsare indexed sets
ofrandom variables; 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 aSchwartz spaceof functions as its
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index set, in which case the continuous random field is
atempered distribution.
We can think about a continuous random field, in a (very)
rough way, as an ordinary function that is almost
everywhere, but such that when we take aweighted
averageof all theinfinitiesover 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 alinear mapfrom a space of functions into
thereal numbers.
Mathematics of fields [edit]
The continuum view (hence the term "field") can be
approached by letting the system have an infinite number
ofdegrees of freedom. The dimension of avector ordinary
differential equationis simply thedimensionof the
vectordependent variable, or thevector function. In this
sensepartial differential equationsso can be thought of as
(coupled)ODEsof infinite dimension (a mathematical
interpretation of the degrees of freedom argument).[15]In
addition, vector fields calledslope fieldsare important tools
in analyzing results in ODEs (see alsophase plane).
The exact nature of the object (and its arguments) in the
differential equation
(e.g.realscalar,complexmatrix,Euclidean vectororfour
vectoretc.) determines the kind of analysis (in our examples
- calculus of a real single variable, a complex matrix and
over real vector fields) needed.
Other than partial differential equations, other parts of
(classical)real analysisandcomplex analysiswere either
inspired by or have techniques applied (or both) in field
theory. Examples of such areas arespectral
theoryandharmonic analysis(vibrations and waves) or the
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