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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: ∇ ⋅ v = 0. {\displaystyle \nabla \cdot \mathbf {v} =0.}
A generalization of this theorem is the Helmholtz decomposition theorem, which states that any vector field can be decomposed as a sum of a solenoidal vector field and an irrotational vector field. By analogy with the Biot-Savart law , A ″ ( x ) {\displaystyle \mathbf {A''} (\mathbf {x} )} also qualifies as a vector potential for v ...
The Helmholtz decomposition in three dimensions was first described in 1849 [9] by George Gabriel Stokes for a theory of diffraction. Hermann von Helmholtz published his paper on some hydrodynamic basic equations in 1858, [10] [11] which was part of his research on the Helmholtz's theorems describing the motion of fluid in the vicinity of vortex lines. [11]
Path independence of the line integral is equivalent to the vector field under the line integral being conservative. A conservative vector field is also irrotational; in three dimensions, this means that it has vanishing curl. An irrotational vector field is necessarily conservative provided that the domain is simply connected.
By its own definition, the vorticity vector is a solenoidal field since = In a two-dimensional flow , ω {\displaystyle {\boldsymbol {\omega }}} is always perpendicular to the plane of the flow, and can therefore be considered a scalar field .
The algorithm of the projection method is based on the Helmholtz decomposition (sometimes called Helmholtz-Hodge decomposition) of any vector field into a solenoidal part and an irrotational part. Typically, the algorithm consists of two stages.
In vector calculus, a complex lamellar vector field is a vector field in three dimensions which is orthogonal to its own curl. [2] That is, =The term lamellar vector field is sometimes used as a synonym for the special case of an irrotational vector field, meaning that [3]
One way to define the curl of a vector field at a point is implicitly through its components along various axes passing through the point: if ^ is any unit vector, the component of the curl of F along the direction ^ may be defined to be the limiting value of a closed line integral in a plane perpendicular to ^ divided by the area enclosed, as ...