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Solenoidal vector field

Vector field with zero divergence

Wikipedia contributorsAdapted for Fathomly · View authors
1 min read
Source updated August 21, 2026
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In vector calculus and differential equations, a divergence free vector field is a vector field v with divergence zero at all points in the field: \nabla \cdot \mathbf {v} =0. In electromagnetic theory, a divergence-free vector field is commonly called a solenoidal vector field, while in hydrodynamics it is commonly called an incompressible vector field, and in mathematical physics it is sometimes called an transverse vector field).

A common way of expressing this property is to say that the field has no sources or sinks, but one must be careful with this interpretation because there are vector fields without sources or sinks that still have non-zero divergence.

01Properties

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:

\;\;\mathbf {v} \cdot \,d\mathbf {S} =0,

where d\mathbf {S} 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 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 subject to certain technical conditions on v, see Helmholtz decomposition.)

02Etymology

Solenoidal has its origin in the Greek word for solenoid, which is σωληνοειδές (sōlēnoeidēs) meaning pipe-shaped, from σωλην (sōlēn) or pipe.

03Examples

Watch videos about Solenoidal vector fieldExplainers and documentaries on YouTube (opens in a new tab)

Sources and credits

This article is adapted from the Wikipedia article “Solenoidal vector field”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Related topics

Helmholtz decomposition

In physics and mathematics, the Helmholtz decomposition theorem or the fundamental theorem of vector calculus states that certain differentiable vector fields can be resolved into the sum of an irrotational vector field and a solenoidal (divergence-free) vector field. In physics, often only the decomposition of sufficiently smooth, rapidly decaying vector fields in three dimensions is discussed.

Stream function

In fluid dynamics, two types of stream function are defined: The two-dimensional (or Lagrange) stream function, introduced by Joseph Louis Lagrange in 1781, is defined for incompressible (divergence-free), two-dimensional flows. The Stokes stream function, named after George Gabriel Stokes, is defined for incompressible, three-dimensional flows with axisymmetry.

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 path between two points does not change the value of the line integral.