solenoidal field
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In
vector calculus Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subjec ...
a solenoidal vector field (also known as an incompressible vector field, a divergence-free vector field, or a transverse vector field) is a vector field v with
divergence In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of t ...
zero at all points in the field: \nabla \cdot \mathbf = 0. A common way of expressing this property is to say that the field has no sources or sinks.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. Zilberti: "The Misconception of Closed Magnetic Flux Lines", IEEE Magnetics Letters, vol. 8, art. 1306005, 2017.


Properties

The divergence theorem 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: where d\mathbf is the outward normal to each surface element. The fundamental theorem of vector calculus states that any vector field can be expressed as the sum of an
irrotational In vector calculus, a conservative vector field is a vector field that is the gradient of some function. A conservative vector field has the property that its line integral is path independent; the choice of any path between two points does not c ...
and a solenoidal field. The condition of zero divergence is satisfied whenever a vector field v has only a
vector potential In vector calculus, a vector potential is a vector field whose curl is a given vector field. This is analogous to a ''scalar potential'', which is a scalar field whose gradient is a given vector field. Formally, given a vector field v, a ''vecto ...
component, because the definition of the vector potential A as: \mathbf = \nabla \times \mathbf automatically results in the identity (as can be shown, for example, using Cartesian coordinates): \nabla \cdot \mathbf = \nabla \cdot (\nabla \times \mathbf) = 0. The
converse Converse may refer to: Mathematics and logic * Converse (logic), the result of reversing the two parts of a definite or implicational statement ** Converse implication, the converse of a material implication ** Converse nonimplication, a logical c ...
also holds: for any solenoidal v there exists a vector potential A such that \mathbf = \nabla \times \mathbf. (Strictly speaking, this holds subject to certain technical conditions on v, see
Helmholtz decomposition In physics and mathematics, in the area of vector calculus, Helmholtz's theorem, also known as the fundamental theorem of vector calculus, states that any sufficiently smooth, rapidly decaying vector field in three dimensions can be resolved into ...
.)


Etymology

''Solenoidal'' has its origin in the Greek word for solenoid, which is σωληνοειδές (sōlēnoeidēs) meaning pipe-shaped, from σωλην (sōlēn) or pipe. In the present context of solenoidal it means constrained as if in a pipe, so with a fixed volume.


Examples

* The magnetic field B (see
Gauss's law for magnetism In physics, Gauss's law for magnetism is one of the four Maxwell's equations that underlie classical electrodynamics. It states that the magnetic field has divergence equal to zero, in other words, that it is a solenoidal vector field. It is ...
) * The
velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
field of an
incompressible fluid flow In fluid mechanics or more generally continuum mechanics, incompressible flow ( isochoric flow) refers to a flow in which the material density is constant within a fluid parcel—an infinitesimal volume that moves with the flow velocity. An eq ...
* The
vorticity In continuum mechanics, vorticity is a pseudovector field that describes the local spinning motion of a continuum near some point (the tendency of something to rotate), as would be seen by an observer located at that point and traveling along wi ...
field * The electric field E in neutral regions (\rho_e = 0); * The current density J where the charge density is unvarying, \frac = 0. * The magnetic vector potential A in Coulomb gauge


See also

*
Longitudinal and transverse vector fields In physics and mathematics, in the area of vector calculus, Helmholtz's theorem, also known as the fundamental theorem of vector calculus, states that any sufficiently smooth, rapidly decaying vector field in three dimensions can be resolved into ...
*
Stream function The stream function is defined for incompressible ( divergence-free) flows in two dimensions – as well as in three dimensions with axisymmetry. The flow velocity components can be expressed as the derivatives of the scalar stream function. T ...
*
Conservative vector field In vector calculus, a conservative vector field is a vector field that is the gradient of some function. A conservative vector field has the property that its line integral is path independent; the choice of any path between two points does not ...


Notes


References

*{{citation , title=Vectors, tensors, and the basic equations of fluid mechanics , authorlink=Rutherford Aris , first=Rutherford , last=Aris , publisher=Dover , year=1989 , isbn=0-486-66110-5 , url=https://books.google.com/books?id=QcZIAwAAQBAJ&q=%22solenoidal+vector+field%22 Vector calculus Fluid dynamics