![Solenoidal vector field 1](https://upload.wikimedia.org/wikipedia/commons/6/69/Solenoidal_vector_field_1.svg)
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 subject ...
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 the ...
zero at all points in the field:
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
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the ''flux'' of a vector field through a closed surface to the ''divergence'' of the field in the vol ...
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
is the outward normal to each surface element.
The
fundamental theorem of vector calculus
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 function, smooth, rapidly decaying vector field in three dimensions can b ...
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:
automatically results in the
identity
Identity may refer to:
* Identity document
* Identity (philosophy)
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* Identity (mathematics)
Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film), ...
(as can be shown, for example, using Cartesian coordinates):
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
(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
upright=1.20, An illustration of a solenoid
upright=1.20, Magnetic field created by a seven-loop solenoid (cross-sectional view) described using field lines
A solenoid () is a type of electromagnet formed by a helix, helical coil of wire whose ...
, 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
A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
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 is a ...
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 ...
* 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 wit ...
field
* The
electric field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...
E in neutral regions (
);
* The
current density
In electromagnetism, current density is the amount of charge per unit time that flows through a unit area of a chosen cross section. The current density vector is defined as a vector whose magnitude is the electric current per cross-sectional ar ...
J where the charge density is unvarying,
.
* The
magnetic vector potential
In classical electromagnetism, magnetic vector potential (often called A) is the vector quantity defined so that its curl is equal to the magnetic field: \nabla \times \mathbf = \mathbf. Together with the electric potential ''φ'', the magnetic v ...
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 flow, incompressible (divergence-free) fluid flow, flows in two dimensions – as well as in three dimensions with axisymmetry. The flow velocity components can be expressed as the derivatives of 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 c ...
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