vector calculus a solenoidal vector field (also known as an incompressible vector field, a divergence-free vector field, or a transverse vector field)...

as the sum of a conservative vector field and a solenoidal field. A line integral of a vector field v {\displaystyle \mathbf {v} } is said to be path-independent...

of the B-field produced this way is that magnetic B-field lines neither start nor end (mathematically, B is a solenoidal vector field); a field line may...

be a solenoidal vector field. Let v : R 3 → R 3 {\displaystyle \mathbf {v} :\mathbb {R} ^{3}\to \mathbb {R} ^{3}} be a solenoidal vector field which...

It states that the magnetic field B has divergence equal to zero, in other words, that it is a solenoidal vector field. It is equivalent to the statement...

solenoidal vector field (one with zero divergence) to S will result in another field that satisfies this required property of a Poynting vector field...

vector field, a vector field on a Riemannian manifold associated with a symmetry Solenoidal vector field, a vector field with zero divergence Vector potential...

calculus identities Vector algebra relations Directional derivative Conservative vector field Solenoidal vector field Laplacian vector field Helmholtz decomposition...

and a solenoidal (divergence-free) vector field. In physics, often only the decomposition of sufficiently smooth, rapidly decaying vector fields in three...

{A}}={\vec {B}}} ( A → {\displaystyle {\vec {A}}} is a vector potential for the solenoidal vector field B → {\displaystyle {\vec {B}}} ). Note that { d F =...

\mathbf {u} =0.} That is, if u {\displaystyle \mathbf {u} } is a solenoidal vector field. A flow is irrotational if the curl of u {\displaystyle \mathbf...

coordinates analysis of solenoidal vector fields, for example, magnetic fields and incompressible fluids. For a three-dimensional vector field F with zero divergence...

(divergence) vanishes, and is called an incompressible flow (sometimes solenoidal vector field). The term incompressible is used because a non-zero divergence...

called solenoidal – in which case any closed surface has no net flux across it. This is the same as saying that the (flow of the) vector field preserves...

outside the sources. In a solenoidal vector field (i.e., a vector field where the divergence is zero everywhere), the field lines neither begin nor end;...

an incompressible fluid the flow velocity field is a solenoidal vector field or a divergence-free vector field. Note that this relationship can be expanded...

magnetic field is a solenoidal vector field. The Maxwell–Faraday version of Faraday's law of induction describes how a time-varying magnetic field corresponds...

the volume form is preserved under the flow of a solenoidal vector field. Thus solenoidal vector fields are precisely those that have volume-preserving...

constant c {\displaystyle \mathbf {c} } or any divergence free (solenoidal) vector field, e.g. Incompressible flows without sources or sinks such as phase...

vectors in other words, ∇ × F = λ F {\displaystyle \nabla \times \mathbf {F} =\lambda \mathbf {F} } . If F {\displaystyle \mathbf {F} } is solenoidal...

Scalar field Solenoidal vector field Stokes' theorem Submersion Surface integral Symmetry of second derivatives Taylor's theorem Total derivative Vector field...

system; also called a starter relay Transmission solenoid Solenoidal vector field, a vector field in mathematics characterized as having zero divergence...

divergence-free field like u {\displaystyle \mathbf {u} } , also known as a solenoidal vector field, can always be represented as the curl of some vector potential...

Compressibility or Incompressible flow a property of a vector field, see Solenoidal vector field a topological property, see Incompressible surface a proof...

aU} The Stokes solution dissipates less energy than any other solenoidal vector field with the same boundary velocities: this is known as the Helmholtz...

velocity field) into the equivalent role of the magnetic induction vector B in electromagnetism. In electromagnetism the B lines form solenoidal rings around...

{\displaystyle \nabla \cdot \mathbf {v} =Q} . For this kind of flow, solenoidal vector fields (no divergence) have no source or sinks. When at a given point...

In mathematics, vector spherical harmonics (VSH) are an extension of the scalar spherical harmonics for use with vector fields. The components of the VSH...

}}\rho \times {\vec {\nabla }}p} This vector, sometimes called the solenoidal vector, is of interest both in compressible fluids and in incompressible...

model generally represent the magnetic field by the solenoidal field B, analogous to the electrostatic field D. A generalization of the above current...