For example, every solenoidal vector field can be written as
2.
Thus solenoidal vector fields are precisely those that have volume-preserving flows.
3.
The solenoidal vector fields are those with div " X " = 0.
4.
It follows from the definition of the Lie derivative that the volume form is preserved under the flow of a solenoidal vector field.
5.
It states that the magnetic field has divergence equal to zero, in other words, that it is a solenoidal vector field.
6.
Equivalent technical statements are that the sum total magnetic flux through any Gaussian surface is zero, or that the magnetic field is a solenoidal vector field.
7.
A generalization of this theorem is the Helmholtz decomposition which states that any vector field can be decomposed as a sum of a solenoidal vector field and an irrotational vector field.
8.
Thinking of a vector field as a 2-form instead, a closed vector field is one whose derivative ( divergence ) vanishes, and is called an incompressible flow ( sometimes solenoidal vector field ).
9.
One important property of the-field produced this way is that magnetic-field lines neither start nor end ( mathematically, is a solenoidal vector field ); a field line either extends to infinity or wraps around to form a closed curve.
10.
This is mathematically equivalent to saying that the divergence of is zero . ( Such vector fields are called solenoidal vector fields . ) This property is called Gauss's law for magnetism and is equivalent to the statement that there are no isolated magnetic poles or magnetic monopoles.
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