Solenoidal vector field

Solenoidal vector field

In vector calculus a solenoidal vector field (also known as an incompressible vector field) is a vector field v with divergence zero at all points in the field;

 \nabla \cdot \mathbf{v} = 0.\,

The fundamental theorem of vector calculus states that any vector field can be expressed as the sum of an irrotational and a solenoidal field. The condition of zero divergence is satisfied whenever a vector field v has only a vector potential component, because the definition of the vector potential A as:

\mathbf{v} = \nabla \times \mathbf{A}

automatically results in the identity (as can be shown, for example, using Cartesian coordinates):

\nabla \cdot \mathbf{v} = \nabla \cdot (\nabla \times \mathbf{A}) = 0.

The converse also holds: for any solenoidal v there exists a vector potential A such that \mathbf{v} = \nabla \times \mathbf{A}. (Strictly speaking, this holds only subject to certain technical conditions on v, see Helmholtz decomposition.)

The divergence theorem, gives the equivalent integral definition of a solenoidal field; namely that for any closed surface S, the net total flux through the surface must be zero:

 \iint_S \mathbf{v} \cdot \, d\mathbf{s} = 0 ,

where d\mathbf{s} is the outward normal to each surface element.

Contents

Etymology

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

Examples

See also

  • Longitudinal and transverse vector fields

References

  • Aris, Rutherford (1989), Vectors, tensors, and the basic equations of fluid mechanics, Dover, ISBN 0486661105 

Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать реферат

Look at other dictionaries:

  • Conservative vector field — In vector calculus a conservative vector field is a vector field which is the gradient of a function, known in this context as a scalar potential. Conservative vector fields have the property that the line integral from one point to another is… …   Wikipedia

  • Lamellar vector field — In vector analysis and in fluid dynamics, a lamellar vector field is a vector field with no rotational component. That is, if the field is denoted as v, then : abla imes mathbf{v} = 0 .A lamellar field is practically synonymous with an… …   Wikipedia

  • 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 negative gradient is a given vector field.Formally, given a vector field v, a… …   Wikipedia

  • Vector calculus — Topics in Calculus Fundamental theorem Limits of functions Continuity Mean value theorem Differential calculus  Derivative Change of variables Implicit differentiation Taylor s theorem Related rates …   Wikipedia

  • Vector spherical harmonics — In mathematics, vector spherical harmonics (VSH) are an extension of the scalar spherical harmonics for the use with vector fields.DefinitionSeveral conventions have been used to define the VSH [R.G. Barrera, G.A. Estévez and J. Giraldo, Vector… …   Wikipedia

  • solenoidal — adjective a) Characteristic of a solenoid b) Describing a vector field having vanishing divergence …   Wiktionary

  • Magnetic field — This article is about a scientific description of the magnetic influence of an electric current or magnetic material. For the physics of magnetic materials, see magnetism. For information about objects that create magnetic fields, see magnet. For …   Wikipedia

  • Maxwell's equations — For thermodynamic relations, see Maxwell relations. Electromagnetism …   Wikipedia

  • Lorentz force — This article is about the equation governing the electromagnetic force. For a qualitative overview of the electromagnetic force, see Electromagnetism. For magnetic force of one magnet on another, see force between magnets. Electromagnetism …   Wikipedia

  • Closed and exact differential forms — In mathematics, especially vector calculus and differential topology, a closed form is a differential form α whose exterior derivative is zero (dα = 0), and an exact form is a differential form that is the exterior derivative of another …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”