F. Xavier Malcata - Mathematics for Enzyme Reaction Kinetics and Reactor Performance

following straightforward algebraic manipulation and condensation afterward. If r is instead set equal to − 1 and y set equal to 1, then Eq. (2.265)gives rise to

(2.272)

– where algebraic rearrangement supports dramatic simplification to

(2.273)

the right‐hand side is but a geometric series of first term equal to 1 and ratio between consecutive terms equal to −x , so one may retrieve Eq. (2.93)to write

(2.274)

since when ∣ x ∣ < 1 – also consistent with ( x + 1) −1representing the reciprocal of x + 1 in the first place, as obtained by long division of 1 by 1 + x following the algorithm depicted in Fig. 2.8. One also finds that

(2.275)

after replacement of x in Eq. (2.265)by its negative, y by 1, and exponent r by a general negative number −z ; Eq. (2.275)eventually yields

(2.276)

If a rising factorial, ( z ) k, is defined as

(2.277)

with evolution opposite to that entailed by Eq. (2.267), one may condense Eq. (2.276)to

(2.278)

In the case of a trinomial, its square may be calculated via

(2.279)

in agreement with Eq. (2.237)applied to x 1 + x 2and x 3, rather than x and y ; a second application of said formula to ( x 1 + x 2) 2generates

(2.280)

that may be rearranged to read

(2.281)

upon elimination of parenthesis. The above reasoning may be applied to any (integer) exponent n , and to any number m of terms of polynomial x 1 + x 2 + ⋯ + x m; the generalized formula looks indeed like

(2.282)

where the summation in the right‐hand side is taken over all sequences of (nonnegative) integer indices k 1through k m, such that the sum of all k i’s is n . The multinomial coefficients are given by

(2.283)

– and count the number of different ways an n ‐element set can be partitioned into disjoint subsets of sizes k 1, k 2, … , k m. For the example above, one would have been led to

(2.284)

after setting m = 3 and n = 2 in Eq. (2.282), while Eq. (2.283)would yield

(2.285)

the possibilities of integer values for ( k 1, k 2, k 3) satisfying the condition placed at the bottom of the summation in Eq. (2.284)encompass (2,0,0), (0,2,0), (0,0,2), (1,1,0), (1,0,1), and (0,1,1) – so Eq. (2.284)becomes

(2.286)

with the aid of Eq. (2.285). After realizing that 0! = 1, 2!/2! = 1, and 2!/1! = 2 – besides – one eventually retrieves Eq. (2.281), using Eq. (2.286)as departure point.

Trigonometry is the branch of mathematics that studies relationships involving lengths of sides and amplitudes of angles in triangles. This field emerged in the Hellenistic world during the third century, by the hand of Euclid and Archimedes – who studied the properties of chords and inscribed angles in circles, while proving theorems equivalent to most modern trigonometric formulae; Hipparchus from Nicaea (Asia Minor) produced, however, the first tables of chords in 140 BCE – analogous to the current tables of sine values, which were completed in the second century CE by Greco‐Egyptian astronomer Ptolemy from Alexandria (Egypt). By those times, it was realized that the lengths of the sides of a right triangle and the angles between those sides satisfy fixed relationships; hence, if at least the length of one side and the amplitude of one angle is known, then all other angles and lengths can be algorithmically determined.

Consider a unit vector u, i.e.

(2.287)

with double bars indicating length, centered at the origin of a system of coordinates, which rotates around said origin – as illustrated in Fig. 2.10a. If the angle defined by vector u(playing the role of hypotenuse in right triangle [ OAB ]) with the horizontal axis is denoted as θ , then cos θ equals, by definition, the ratio of the length of the adjacent leg, [ OA ], to the length of the hypotenuse, [ OB ], according to