612
B Review of Traditional Calculus
dF
dT
=
d
dT
b(T )
c
f (t, T ) dt −
a(T )
c
f (t, T ) dt
,
(B.1.28)
then employing the chain rule version of the fundamental theorem to give
dF
dT
=
∂
∂b
b
c
f (t, T ) dt
db
dT
+
∂
∂T
b
c
f (t, T ) dt
−
∂
∂a
a
c
f (t, T ) dt
da
dT
−
∂
∂T
a
c
f (t, T ) dt ,
(B.1.29)
which can be simplified to give the final Leibniz rule as
d
dT
b(T )
a(T )
f (t, T ) dt =
b(T )
a(T )
∂f
∂T
dt +
db
dT
f [b(T ), T ] −
da
dT
f [(a(T ), T ] .
(B.1.30)
B.1.7 Regression of Series Concept
Let us consider that we may express a quantity x as a power series in the variable y,
viz.,
x =
∞
i=1
a i y
i ,
(B.1.31)
in which the coefficients a i are known. However, if what we would actually like
to have is y expressed as a power series in x, how do we go about obtaining the
appropriate expression? We shall have occasion to ask this question a couple of
times in the course of our study of statistical mechanics.
To answer this query, we shall begin by assuming that y can be written as a power
series in x, namely,
y ≡
∞
j =1
b j x
j ,
(B.1.32)
in which the coefficients b j are for the moment unknown (and hence to be
determined). To see how this procedure works, we shall obtain expressions for b 1 ,
b 2 , b 3 , and b 4 by substituting Eq. (B.1.32) into Eq. (B.1.31), followed by collecting
like powers of x on the right-hand side of the equation. This procedure results in x
being expressed as
B Review of Traditional Calculus
dF
dT
=
d
dT
b(T )
c
f (t, T ) dt −
a(T )
c
f (t, T ) dt
,
(B.1.28)
then employing the chain rule version of the fundamental theorem to give
dF
dT
=
∂
∂b
b
c
f (t, T ) dt
db
dT
+
∂
∂T
b
c
f (t, T ) dt
−
∂
∂a
a
c
f (t, T ) dt
da
dT
−
∂
∂T
a
c
f (t, T ) dt ,
(B.1.29)
which can be simplified to give the final Leibniz rule as
d
dT
b(T )
a(T )
f (t, T ) dt =
b(T )
a(T )
∂f
∂T
dt +
db
dT
f [b(T ), T ] −
da
dT
f [(a(T ), T ] .
(B.1.30)
B.1.7 Regression of Series Concept
Let us consider that we may express a quantity x as a power series in the variable y,
viz.,
x =
∞
i=1
a i y
i ,
(B.1.31)
in which the coefficients a i are known. However, if what we would actually like
to have is y expressed as a power series in x, how do we go about obtaining the
appropriate expression? We shall have occasion to ask this question a couple of
times in the course of our study of statistical mechanics.
To answer this query, we shall begin by assuming that y can be written as a power
series in x, namely,
y ≡
∞
j =1
b j x
j ,
(B.1.32)
in which the coefficients b j are for the moment unknown (and hence to be
determined). To see how this procedure works, we shall obtain expressions for b 1 ,
b 2 , b 3 , and b 4 by substituting Eq. (B.1.32) into Eq. (B.1.31), followed by collecting
like powers of x on the right-hand side of the equation. This procedure results in x
being expressed as