Divergenceless Vector Field

A simple proof is given of the theorem that an arbitrary divergenceless vector field F can be represented in terms of two scalar fields Debye potentials and according to FLL, where L is the angular momentum operator.

So for any vector eld G and any function f, curlG curlG f, i.e, we can change the vector eld G in a controllable way without changing its curl. This allows us to simplify the task of nding G by rst choosing a function f with f z U e.g., integrate U, dz!, so Gf S f x,Tf y,Uf z S f x,Tf y,0 and this has

An example of a solenoidal vector field, , , 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 A common way of expressing this property is to say that the field has no sources or sinks.

If it is known that a vector field 92vec V is divergence free, that is, 92nabla 92cdot 92vec V 0 which by the fundamental theorem of vector calculus implies the field may be expressed as, 92vec V 92nabla 92times 92vec A

In physics one often meets the criterion that a vector field has zero divergence, e.g. 92nabla92cdot92mathbfv 0 92quad 1 Based on this one often claim that the vector field can be written as the curl of another field, 92mathbfv92nabla92times92mathbfA 92quad 2 since 92nabla92cdot 92nabla 92times 92mathbff 0 for any smooth 92mathbf

from which 92192Rightarrow392 follows immediately. To show 92392Rightarrow192text,92 one must use the fact that the right-hand side of this equation now vanishes by assumption for any region whose boundary is the given surface, which forces the integrand, and not merely the integral, on the left-hand side to vanish.

Furthermore, since F must be the curl of some vector field A, we can further conclude that tex92int_92partial V 92nabla 92times A 92cdot dA 0tex or in other words, that the surface integral of the curl of the vector function A over a closed surface must be zero. Similar threads.

A divergenceless vector field, also called a solenoidal field, is a vector field for which del F0. Therefore, there exists a G such that Fdel xG. Furthermore, F can be written as F del xTrdel 2Sr 1 TS, 2 where T del xTr 3 -rxdel T 4 S del 2Sr 5 del partialpartialrrS-rdel 2S. 6 Following Lamb's 1932 treatise Lamb 1993, T and S are called

STREAM FUNCTIONS FOR DIVERGENCE-FREE VECTOR FIELDS JAMES P. KELLIHER1 Abstract. In 1990, Von Wahl and, independently, Borchers and Sohr showed that a divergence-free vector eld uin a 3D bounded domain that is tangential to the bound-ary can be written as the curl of a vector eld vanishing on the boundary of the domain.

By the Lie symmetry group, the reduction for divergence-free vector-fields DFVs is studied, and the following results are found. A n-dimensional DFV can be locally reduced to a n 1-dimensional DFV if it admits a one-parameter symmetry group that is spatial and divergenceless. More generally, a n-dimensional DFV admitting a r-parameter, spatial, divergenceless Abelian commutable