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

Although seldom of relevance, higher order constants, say A l, 3, A l, 4,…, may be calculated in a similar fashion – by resorting to higher order derivatives of Eq. (2.220), but accordingly requiring more cumbersome algebraic manipulations.

According to Newton’s theorem, it is possible to expand the n th power (with n integer) of a sum of (real) terms, x and y , via a sum of a finite number of products of integer powers of x and y , with exponents adding up to n in every case; more specifically,

(2.236)

Equation (2.236)is also known as binomial formula, or binomial identity – and x and y may represent numbers, or instead functions. The first description of the above formula in its entirety, for nonnegative integer n , was due to Blaise Pascal – a French physicist, mathematician, and philosopher who lived in the seventeenth century; while Greek mathematician Euclid referred to its second order form in the fourth century BCE, and Indian mathematician Pingala mentioned higher orders one century later. However, it was Isaac Newton – who extended it to every real exponent in 1665, that has received credit for such a relationship thereafter.

The simplest case pertains indeed to n = 2, and accordingly reads

(2.237)

– being also known as another notable case of multiplication; for positive values of x and y , this can be graphically illustrated as in Fig. 2.9. The area ( x + y ) 2of the larger square of side x + y may indeed be obtained via addition of the area of two smaller squares of sides x and y , i.e. x 2and y 2, respectively, to the area xy of each of two rectangles of sides x and y .

Figure 2.9 Geometric demonstration of Newton’s binomial formula at two dimensions – resorting to squares of side x and area x 2, side y and area y 2, and side x + y and area ( x + y ) 2, complemented with two rectangles of sides x and y and area xy .

If y is replaced by −y , then Eq. (2.237)gives rise to

(2.238)

– also known as another notable case of multiplication; however, a simple geometrical interpretation is no longer possible (due to the negative term). A similar proof may be constructed in three dimensions, corresponding to the volume ( x + y ) 3of a cube of side x + y ; it may indeed be decomposed as

(2.239)

i.e. the sum of volume x 2 y of each of three parallelipipeds of sides x , x , and y , to the volume xy 2of each of three parallelipipeds of sides x , y , and y , and finally to the volume y 3of a cube of side y – besides being directly obtainable from Eq. (2.236)after setting n = 3.

The binomial coefficients in Eq. (2.236), of the form , count in how many ways one can pick up a subset with k elements out from a set of n elements in total; in mathematical terms, this is equivalent to writing

(2.240)

and supports the entries of Pascal’s triangle – denoted as Table 2.1. Careful inspection of this table indicates that the outermost values are always unity, whereas every two consecutive numbers in a given row add up to the value placed in between at the next row. In fact, Eq. (2.240)allows one to write

(2.241)

where factoring n ! coupled with elimination of inner parenthesis give rise to

(2.242)

the factorials in denominator may, in turn, be rewritten as

(2.243)

based on their definition, thus allowing further factoring out of ( k − 1)! and ( n − k )! as

(2.244)

Upon lumping the two factors still in parenthesis, Eq. (2.244)becomes

(2.245)

that degenerates to

(2.246)

the outstanding factors may, in turn, be lumped with the existing factorials to yield

(2.247)

– equivalent, in view of Eq. (2.240), to

(2.248)

thus confirming the initial suggestion and being frequently known as Pascal’s rule.

Table 2.1 Pascal’s triangle encompassing coefficients of power of binomial, , for the first values of n – and, in each case, for k = 0, 1, … , n − 1 , n .

In order to exactly prove Eq. (2.236), one should start by realizing that