20:
169:
629:
This statement does not mean that the field lines of a solenoidal field must be closed, neither that they cannot begin or end. For a detailed discussion of the subject, see J. Slepian: "Lines of Force in
Electric and Magnetic Fields", American Journal of Physics, vol. 19, pp. 87-90, 1951, and L.
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79:
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gives an equivalent integral definition of a solenoidal field; namely that for any closed surface, the net total flux through the surface must be zero:
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382:
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534:
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Zilberti: "The
Misconception of Closed Magnetic Flux Lines", IEEE Magnetics Letters, vol. 8, art. 1306005, 2017.
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and a solenoidal field. The condition of zero divergence is satisfied whenever a vector field
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A common way of expressing this property is to say that the field has no sources or sinks.
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441:, which is σωληνοειδές (sōlēnoeidēs) meaning pipe-shaped, from σωλην (sōlēn) or pipe.
360:{\displaystyle \nabla \cdot \mathbf {v} =\nabla \cdot (\nabla \times \mathbf {A} )=0.}
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252:
107:
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418:(Strictly speaking, this holds subject to certain technical conditions on
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states that any vector field can be expressed as the sum of an
307:(as can be shown, for example, using Cartesian coordinates):
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Vectors, tensors, and the basic equations of fluid mechanics
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component, because the definition of the vector potential
411:{\displaystyle \mathbf {v} =\nabla \times \mathbf {A} .}
209:{\displaystyle \;\;\mathbf {v} \cdot \,d\mathbf {S} =0,}
574:{\textstyle {\frac {\partial \rho _{e}}{\partial t}}=0}
296:{\displaystyle \mathbf {v} =\nabla \times \mathbf {A} }
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244:is the outward normal to each surface element.
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144:{\displaystyle \nabla \cdot \mathbf {v} =0.}
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602:Longitudinal and transverse vector fields
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23:An example of a solenoidal vector field,
74:{\displaystyle \mathbf {v} (x,y)=(y,-x)}
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531:where the charge density is unvarying,
249:fundamental theorem of vector calculus
437:has its origin in the Greek word for
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117:zero at all points in the field:
16:Vector field with zero divergence
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375:there exists a vector potential
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371:also holds: for any solenoidal
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303:automatically results in the
237:{\displaystyle d\mathbf {S} }
98:divergence-free vector field
514:{\displaystyle \rho _{e}=0}
94:incompressible vector field
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612:Conservative vector field
586:magnetic vector potential
470:incompressible fluid flow
459:Gauss's law for magnetism
424:Helmholtz decomposition
103:transverse vector field
90:solenoidal vector field
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161:divergence theorem
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92:(also known as an
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253:irrotational
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108:vector field
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259:has only a
670:Categories
637:References
435:Solenoidal
379:such that
155:Properties
115:divergence
652:, Dover,
557:∂
546:ρ
542:∂
497:ρ
476:vorticity
430:Etymology
398:×
395:∇
341:×
338:∇
332:⋅
329:∇
318:⋅
315:∇
286:×
283:∇
186:⋅
128:⋅
125:∇
63:−
646:(1989),
596:See also
466:velocity
445:Examples
439:solenoid
369:converse
305:identity
106:) is a
100:, or a
656:
422:, see
219:where
618:Notes
478:field
457:(see
113:with
654:ISBN
584:The
524:The
481:The
474:The
464:The
450:The
367:The
267:as:
247:The
159:The
96:, a
426:.)
84:In
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355:0.
139:0.
88:a
589:A
581:.
569:0
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392:=
388:v
377:A
373:v
352:=
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345:A
335:(
326:=
322:v
290:A
280:=
276:v
265:A
257:v
231:S
227:d
204:,
201:0
198:=
194:S
190:d
182:v
136:=
132:v
111:v
69:)
66:x
60:,
57:y
54:(
51:=
48:)
45:y
42:,
39:x
36:(
32:v
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