Source Word
Legendre transformation
Noun
(mathematics)
Given
a
function
f(x,y,z,...)
which
is
concave
up
with
respect
to
x
(i.e.,
its
second
derivative
with
respect
to
x
is
greater
than
zero),
an
involutive
procedure
for
replacing
x
with
another
variable,
say
p=∂f/∂x
thus
yielding
another
function,
say
F=F(p,y,z,...).
This
new
function
contains
all
of
the
information
of
the
original
f
encoded,
as
it
were,
within
it
so
that
∂F/∂p=x
and
applying
a
similar
transformation
to
F
yields
the
original
f.
The
formula
is:
F(p,y,z,...)=p·x(p)-f(x(p),y,z,...)
where
x
must
be
expressed
as
a
function
of
p.
(Note:
The
concave
upwardness
means
that
∂f/∂x
is
monotonically
increasing,
which
means
that
p
as
a
function
of
x
is
invertible,
so
x
should
be
expressible
as
a
function
of
p.)
Japanese Meaning
数学におけるルジャンドル変換とは、ある関数 f(x,y,z,...) が変数 x に関して上に凸である（すなわち、x に関する二階微分が正である）場合に、x をその偏微分 p = ∂f/∂x により新たな変数 p と置き換え、F(p,y,z,...) = p·x(p) - f(x(p),y,z,...) という形で定義される新しい関数に変換する手法です。この変換により、元の関数 f の全ての情報が F に含まれることになります。
Sense（1）
(mathematics)
Given
a
function
f(x,y,z,...)
which
is
concave
up
with
respect
to
x
(i.e.,
its
second
derivative
with
respect
to
x
is
greater
than
zero),
an
involutive
procedure
for
replacing
x
with
another
variable,
say
p=∂f/∂x
thus
yielding
another
function,
say
F=F(p,y,z,...).
This
new
function
contains
all
of
the
information
of
the
original
f
encoded,
as
it
were,
within
it
so
that
∂F/∂p=x
and
applying
a
similar
transformation
to
F
yields
the
original
f.
The
formula
is:
F(p,y,z,...)=p·x(p)-f(x(p),y,z,...)
where
x
must
be
expressed
as
a
function
of
p.
(Note:
The
concave
upwardness
means
that
∂f/∂x
is
monotonically
increasing,
which
means
that
p
as
a
function
of
x
is
invertible,
so
x
should
be
expressible
as
a
function
of
p.)
Sense（2）
Sense（3）
(thermodynamics)
A
relation
between
internal
energy
(expressed
in
terms
of
volume
and
entropy)
and
enthalpy
(replacing
volume
with
pressure),
or
between
internal
energy
and
Helmholtz
free
energy
(replacing
entropy
with
temperature),
or
between
enthalpy
and
Gibbs
free
energy
(replacing
entropy
with
temperature),
or
between
internal
energy
and
Gibbs
free
energy
(replacing
volume
with
pressure
and
entropy
with
temperature),
or
between
Helmholtz
free
energy
and
Gibbs
free
energy
(replacing
volume
with
pressure).
（ plural ）
