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...

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...

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

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...

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...

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

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

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

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

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;...

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

\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...

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...

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

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...

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...

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...

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

magnetic fields are different facets of the same electromagnetic field, and in moving from one inertial frame to another, the solenoidal vector field portion...

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

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

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

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

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

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

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...

\mathbf {a} } (such as a magnetic field) is being advected by the solenoidal velocity field u {\displaystyle \mathbf {u} } , then the advection equation above...