B.1 Derivatives
611
n
i=0
f (i) =
n
0
f (t) dt +
1
2 [f (0) + f (n)] −
∞
k=1
B 2k
(2k)!
f
(2k−1) (0) − f
(2k−1) (n)
.
(B.1.23)
This result is referred to as the Euler–Maclaurin summation formula.
B.1.6 Fundamental Theorem of the Calculus
The fundamental theorem of calculus, as expressed in first year calculus courses, is
d
dx
x
a
f (t) dt = f (x) .
(B.1.24)
The fundamental theorem can be generalized simply to deal with composite
functions by utilizing the chain rule for derivatives, as follows:
d
dx
g(x)
a
f (t) dt =
dg
dx
d
dg
g
a
f (t) dt = f (g(x))
dg
dx
.
(B.1.25)
We sometimes encounter functions in statistical thermodynamics that are defined
according to this theorem. You will see this in particular in the Debye model for the
heat capacity of a solid.
There will also be occasion for us to utilize a result, known as the Leibniz
parameter differentiation rule, that arises when we consider functions of the form
F (T ) =
b(T )
a(T )
f (t, T ) dt ,
(B.1.26)
in which the upper and/or lower limit of the integral is a function of the parameter
(or variable) for which the derivative is needed. What is its derivative with respect
to T ? Let us work it out. Formally, it can be written as
dF
dT
=
d
dT
b(T )
a(T )
f (t, T ) dt .
(B.1.27)
What we need to do to proceed with evaluating this derivative is to convert our
expression into one that will allow us to utilize the fundamental theorem of the
calculus. We may do this by introducing a constant c into the limit of the integral as
follows:
into the higher-derivatives summation term and employing a redefined set of Bernoulli numbers
that are solely positive.
611
n
i=0
f (i) =
n
0
f (t) dt +
1
2 [f (0) + f (n)] −
∞
k=1
B 2k
(2k)!
f
(2k−1) (0) − f
(2k−1) (n)
.
(B.1.23)
This result is referred to as the Euler–Maclaurin summation formula.
B.1.6 Fundamental Theorem of the Calculus
The fundamental theorem of calculus, as expressed in first year calculus courses, is
d
dx
x
a
f (t) dt = f (x) .
(B.1.24)
The fundamental theorem can be generalized simply to deal with composite
functions by utilizing the chain rule for derivatives, as follows:
d
dx
g(x)
a
f (t) dt =
dg
dx
d
dg
g
a
f (t) dt = f (g(x))
dg
dx
.
(B.1.25)
We sometimes encounter functions in statistical thermodynamics that are defined
according to this theorem. You will see this in particular in the Debye model for the
heat capacity of a solid.
There will also be occasion for us to utilize a result, known as the Leibniz
parameter differentiation rule, that arises when we consider functions of the form
F (T ) =
b(T )
a(T )
f (t, T ) dt ,
(B.1.26)
in which the upper and/or lower limit of the integral is a function of the parameter
(or variable) for which the derivative is needed. What is its derivative with respect
to T ? Let us work it out. Formally, it can be written as
dF
dT
=
d
dT
b(T )
a(T )
f (t, T ) dt .
(B.1.27)
What we need to do to proceed with evaluating this derivative is to convert our
expression into one that will allow us to utilize the fundamental theorem of the
calculus. We may do this by introducing a constant c into the limit of the integral as
follows:
into the higher-derivatives summation term and employing a redefined set of Bernoulli numbers
that are solely positive.