Solenoidal vector field

 ( Vector Calculus ) 

In vector calculus 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 zero at all points in the field: $$\backslash nabla\; \backslash cdot\; \backslash mathbf\{v\}\; =\; 0.$$ 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 $d\backslash mathbf\{S\}$ 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: $$\backslash mathbf\{v\}\; =\; \backslash nabla\; \backslash times\; \backslash mathbf\{A\}$$ automatically results in the identity (as can be shown, for example, using Cartesian coordinates): $$\backslash nabla\; \backslash cdot\; \backslash mathbf\{v\}\; =\; \backslash nabla\; \backslash cdot\; (\backslash nabla\; \backslash times\; \backslash mathbf\{A\})\; =\; 0.$$ The converse also holds: for any solenoidal v there exists a vector potential A such that $\backslash mathbf\{v\}\; =\; \backslash nabla\; \backslash times\; \backslash mathbf\{A\}.$ (Strictly speaking, this holds subject to certain technical conditions on v, see Helmholtz decomposition.)

---

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

---

Examples

• The magnetic field B (see Gauss's law for magnetism)
• The velocity field of an incompressible fluid flow
• The vorticity field
• The electric field E in neutral regions ($\backslash rho\_e\; =\; 0$);
• The current density J where the charge density is unvarying, $\backslash frac\{\backslash partial\; \backslash rho\_e\}\{\backslash partial\; t\}\; =\; 0$.
• The magnetic vector potential A in Coulomb gauge

---

See also

• Longitudinal and transverse vector fields
• Stream function
• Conservative vector field

---

Notes

Post Comment