B.2 Legendre Transforms
615
We note in passing that the expression y(x) = x 2 + 2x + y 0 represents a family
of parabolas, all having a common derivative function y (x). In other words, y(x)
is the general solution to the first-order linear ordinary differential equation y (x) =
2x + 2, and y 0 thus represents the constant of integration for its solution y(x) =
y dx + y 0 . Notice also, that for the specific example with y 0 = 3, we may solve
y (x) = 2x + 2 for x to obtain x(y ) =
1
2 y − 1 which, upon substitution into y(x),
gives y as a function of y as y(y ) =
1
4 (y ) 2 + 2. Alternatively, for the present
example, we could explicitly invert y(x) = x 2 + 2x + 3 by solving the quadratic
equation x 2 + 2x + 3 − y = 0 for x to obtain x = −1 ±
√
y − 2, which corresponds
to two separate functions of y on the domain y ∈ [3, ∞), since for x to qualify as a
function of y, it must be single-valued on its domain of definition. You might wish
to check that this also gives the same result for y(y ).
We see from Fig. B.1 that the set of tangent lines is indeed equivalent to the
curve mapped out by the set of points (x, y) in the sense that y = y(x) represents
the envelope of the family of tangent lines. This illustrates what is referred to as the
duality between the point geometry with which we are most familiar and the line
geometry introduced by Plücker in the nineteenth century. This duality may also
be expressed by noting that in the more traditional point geometry representation,
the relevant information is conveyed by the set of points (x, y), with y given by an
expression y = y(x), while in the Plücker line geometry the same information is
conveyed by a set of points (y , t i ) giving the slopes and y-intercepts of the set of
tangent lines whose envelope corresponds to the expression y = y(x).
Nothing that has been said here precludes us from extending the Legendre
transform concept to multivariable functions. As discussed by Callen, 6 z = f (x, y)
represents a surface that can be viewed as being mapped out by the set of points
(x, y, z) or as the envelope of the set of tangent planes to the surface. The most
general Legendre transform for z would then be
t i (x) = z −
∂f
∂x
y
x −
∂f
∂y
x
y ,
(B.2.2b)
involving both independent variables x and y. Note, however, that we may still carry
out a partial Legendre transform that involves only x or only y, while leaving the
other variable unchanged.
The total differential df for a function of two variables f ≡ f (x, y) is given as
df = u dx + v dy ;
u ≡
∂f
∂x
y
; v ≡
∂f
∂y
x
.
(B.2.3)
6 H. B. Callen, Thermodynamics and an Introduction to Thermostatics, 2nd Edition (Wiley, New
York, 1985), §§5.2, 5.3, and 12.4.
615
We note in passing that the expression y(x) = x 2 + 2x + y 0 represents a family
of parabolas, all having a common derivative function y (x). In other words, y(x)
is the general solution to the first-order linear ordinary differential equation y (x) =
2x + 2, and y 0 thus represents the constant of integration for its solution y(x) =
y dx + y 0 . Notice also, that for the specific example with y 0 = 3, we may solve
y (x) = 2x + 2 for x to obtain x(y ) =
1
2 y − 1 which, upon substitution into y(x),
gives y as a function of y as y(y ) =
1
4 (y ) 2 + 2. Alternatively, for the present
example, we could explicitly invert y(x) = x 2 + 2x + 3 by solving the quadratic
equation x 2 + 2x + 3 − y = 0 for x to obtain x = −1 ±
√
y − 2, which corresponds
to two separate functions of y on the domain y ∈ [3, ∞), since for x to qualify as a
function of y, it must be single-valued on its domain of definition. You might wish
to check that this also gives the same result for y(y ).
We see from Fig. B.1 that the set of tangent lines is indeed equivalent to the
curve mapped out by the set of points (x, y) in the sense that y = y(x) represents
the envelope of the family of tangent lines. This illustrates what is referred to as the
duality between the point geometry with which we are most familiar and the line
geometry introduced by Plücker in the nineteenth century. This duality may also
be expressed by noting that in the more traditional point geometry representation,
the relevant information is conveyed by the set of points (x, y), with y given by an
expression y = y(x), while in the Plücker line geometry the same information is
conveyed by a set of points (y , t i ) giving the slopes and y-intercepts of the set of
tangent lines whose envelope corresponds to the expression y = y(x).
Nothing that has been said here precludes us from extending the Legendre
transform concept to multivariable functions. As discussed by Callen, 6 z = f (x, y)
represents a surface that can be viewed as being mapped out by the set of points
(x, y, z) or as the envelope of the set of tangent planes to the surface. The most
general Legendre transform for z would then be
t i (x) = z −
∂f
∂x
y
x −
∂f
∂y
x
y ,
(B.2.2b)
involving both independent variables x and y. Note, however, that we may still carry
out a partial Legendre transform that involves only x or only y, while leaving the
other variable unchanged.
The total differential df for a function of two variables f ≡ f (x, y) is given as
df = u dx + v dy ;
u ≡
∂f
∂x
y
; v ≡
∂f
∂y
x
.
(B.2.3)
6 H. B. Callen, Thermodynamics and an Introduction to Thermostatics, 2nd Edition (Wiley, New
York, 1985), §§5.2, 5.3, and 12.4.