Helmholtz's theorems

Three results in ideal-fluid mechanics state that vortex lines move with the fluid, vortex strength is conserved, and vortex tubes cannot end inside the fluid.

Vorticity equation: An evolution equation describing how a fluid's vorticity changes under flow, stretching, and other forces. Its ideal-fluid form yields the transport and conservation statements in Helmholtz's theorems.
Vorticity: The local angular rotation of a fluid, represented by the curl of its velocity field. Helmholtz's theorems concern the geometry and evolution of this field.
Hermann von Helmholtz: A German physicist and physician whose work shaped acoustics, electrodynamics, and fluid mechanics. He formulated the classic vortex theorems in his 1858 study of discontinuous fluid motion.
Navier–Stokes equations: Partial differential equations governing fluid motion with viscosity and other forces. Viscous terms can diffuse or destroy vorticity, unlike the idealized theorem setting.
Kelvin's circulation theorem: A theorem stating that circulation around a material loop is conserved in an inviscid barotropic flow with conservative body forces. Conserved circulation underlies the conservation of vortex strength.
Circulation: The line integral of fluid velocity around a closed curve. Circulation provides the measure connected to conserved vortex strength.
On Integrals of the Hydrodynamical Equations, Which Express Vortex-motion: Helmholtz's 1858 paper deriving foundational results about vortex motion in fluids. This paper introduced the theorems associated with his name.
Baroclinic vorticity generation: The production of vorticity when pressure and density gradients are not parallel. This mechanism changes circulation in non-barotropic flows, outside a key theorem condition.
Vortex line: A curve everywhere tangent to the vorticity vector at a given instant in a fluid. The first theorem says these curves move with the fluid particles.
Inviscid flow: Fluid motion modeled without viscous stresses. The theorems assume idealized motion without viscosity.

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Helmholtz decompositionRelated: It is closely related, but emphasizes reconstruction and uniqueness rather than the decomposition itself.