LIE ALGEBRA OF DIVERGENCE-FREE VECTOR FIELDS AND SOLENOIDALITY OF KILLING VECTOR FIELDS
The article investigates solenoidal vector fields and demonstrates that they form
a Lie algebra with respect to their Lie bracket, while also proving that Killing
vector fields are solenoidal. These fields are pivotal in the Helmholtz decomposition
theorem and have diverse applications, such as in the design of solenoid valves
and electromagnets. Furthermore, the proven theorems confirm that the space of
solenoidal vector fields constitutes a Lie algebra, with the Lie bracket acting as the
multiplication operation.
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