Computation and Properties of Maps
131
An interesting side aspect is that with the search for a multiplicative inverse
in eq. (5.14) one has derived a rule to differentiate the function f (x) = 1/x
without explicitly using calculus rules.
After discussing the algebra 1 D 1 and its virtues for the computation of
derivatives, we now address the most general Differential Algebra, the structure n D v , which corresponds to the case of power series of v variables to
the nth order. It will eventually allow us to arithmetically compute partial
derivatives of functions of v variables through order n.
5.2.2 The Structure n D v
We define N (n, v) to be the number of monomials in v variables through
order n. We will show that
N (n, v) =
(n + v)!
n!v!
= C(n + v, v),
where C(i, j) is the familiar binomial coefficient. First note that the number
of monomials with exact order n equals N (n, v − 1). This is true because each
monomial of exact order n can be written as a monomial with one variable
less times the last variable to such a power that the total power equals n.
Thus we have
N (n, v) = N (n − 1, v) + N (n, v − 1) :
the number of monomials in v variables through order n equals the number
of monomials of one order less plus the ones of exact order n. This recursive
relation is satisfied by C(n + v, v). Since also obviously C(1 + 1, 1) = 2 =
N (1, 1), the formula follows by induction.
Now assume that all these N monomials are arranged in a certain manner
order-by-order. For each monomial M we call I M the position of M according
to the ordering. Conversely, with M I we denote the Ith monomial of the
ordering. Finally, for an I with M I = x
i1
1 · · · x
iv
v we define F I = i 1 ! · · · i v !.
We now define an addition, a scalar multiplication and a vector multiplication on R
N in the following way:
(a 1 , . . . , a N ) + (b 1 , . . . , b N ) = (a 1 + b 1 , · · · , a N + b N ),
t · (a 1 , . . . , a N ) = (t · a 1 , · · · , t · a N ),
(a 1 , . . . , a N ) · (b 1 , . . . , b N ) = (c 1 , . . . , c N ),
(5.17)
where the coefficients c i are defined as follows:
c i = F i
0≤ν,μ≤N
Mν ·Mμ=Mi
a ν · b μ
F ν · F μ
.
(5.18)
To help clarify these definitions, let us look at the case of two variables
and second order. In this case, we have n = 2 and v = 2. There are N =
131
An interesting side aspect is that with the search for a multiplicative inverse
in eq. (5.14) one has derived a rule to differentiate the function f (x) = 1/x
without explicitly using calculus rules.
After discussing the algebra 1 D 1 and its virtues for the computation of
derivatives, we now address the most general Differential Algebra, the structure n D v , which corresponds to the case of power series of v variables to
the nth order. It will eventually allow us to arithmetically compute partial
derivatives of functions of v variables through order n.
5.2.2 The Structure n D v
We define N (n, v) to be the number of monomials in v variables through
order n. We will show that
N (n, v) =
(n + v)!
n!v!
= C(n + v, v),
where C(i, j) is the familiar binomial coefficient. First note that the number
of monomials with exact order n equals N (n, v − 1). This is true because each
monomial of exact order n can be written as a monomial with one variable
less times the last variable to such a power that the total power equals n.
Thus we have
N (n, v) = N (n − 1, v) + N (n, v − 1) :
the number of monomials in v variables through order n equals the number
of monomials of one order less plus the ones of exact order n. This recursive
relation is satisfied by C(n + v, v). Since also obviously C(1 + 1, 1) = 2 =
N (1, 1), the formula follows by induction.
Now assume that all these N monomials are arranged in a certain manner
order-by-order. For each monomial M we call I M the position of M according
to the ordering. Conversely, with M I we denote the Ith monomial of the
ordering. Finally, for an I with M I = x
i1
1 · · · x
iv
v we define F I = i 1 ! · · · i v !.
We now define an addition, a scalar multiplication and a vector multiplication on R
N in the following way:
(a 1 , . . . , a N ) + (b 1 , . . . , b N ) = (a 1 + b 1 , · · · , a N + b N ),
t · (a 1 , . . . , a N ) = (t · a 1 , · · · , t · a N ),
(a 1 , . . . , a N ) · (b 1 , . . . , b N ) = (c 1 , . . . , c N ),
(5.17)
where the coefficients c i are defined as follows:
c i = F i
0≤ν,μ≤N
Mν ·Mμ=Mi
a ν · b μ
F ν · F μ
.
(5.18)
To help clarify these definitions, let us look at the case of two variables
and second order. In this case, we have n = 2 and v = 2. There are N =