Last Updated：2025/12/31
Sentence
In
the
language
of
differential
geometry,
the
incompressible
inviscid
Euler
equations
can
be
written
in
vorticity-vector
potential
form
as
∂ₜ𝜔+ℒᵤ𝜔=0\u=𝛿̃𝜂⁻¹𝛥⁻¹𝜔
where
𝜔
is
the
vorticity
2-form,
ℒᵤ
denotes
the
Lie
derivative
with
respect
to
the
velocity
field
u,
𝛥
is
the
Hodge
Laplacian,
𝛿
is
the
codifferential
(the
negative
of
the
divergence
operator),
and
̃𝜂⁻¹
is
the
canonical
map
from
2-forms
to
2-vector
fields
induced
by
the
Euclidean
metric
𝜂.
Quizzes for review
In the language of differential geometry, the incompressible inviscid Euler equations can be written in vorticity-vector potential form as ∂ₜ𝜔+ℒᵤ𝜔=0\u=𝛿̃𝜂⁻¹𝛥⁻¹𝜔 where 𝜔 is the vorticity 2-form, ℒᵤ denotes the Lie derivative with respect to the velocity field u, 𝛥 is the Hodge Laplacian, 𝛿 is the codifferential (the negative of the divergence operator), and ̃𝜂⁻¹ is the canonical map from 2-forms to 2-vector fields induced by the Euclidean metric 𝜂.
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