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

so cotanh −1 x tends to (positive) infinite when x = 1 is approached – meaning that x = 1 serves as vertical asymptote as well. For the remainder of its domain, this inverse function is monotonically decreasing in either interval] − ∞, − 1[ or]1,∞[; when x →−∞, one obtains

(2.603)

stemming from Eq. (2.600)– and similarly when x → ∞, i.e.

(2.604)

because x + 1 ≈ x and x − 1 ≈ x , thus indicating that the horizontal axis plays the role of (single) horizontal asymptote when x grows unbounded.

As indicated previously, a vector uis defined as a quantity possessing both a magnitude and a direction; the said magnitude is regularly denoted by ‖ u‖, while information on the direction is often conveyed graphically – or else encompasses angles formed with the axes in some reference system. Two vectors, uand v, are said to be equal when their magnitudes are identical, i.e. ‖ u‖ = ‖ v‖, and also point in the same direction; however, they do not need to have the same origin.

A much more convenient way of handling vectors resorts, however, to their decomposition along the three directions of space in a typical Cartesian R 3domain, according to

(3.1)

and

(3.2)

here j x, j y, and j zdenote unit, orthogonal vectors of a Cartesian system, defined as

(3.3)

(3.4)

and

(3.5)

– while

(3.6)

and

(3.7)

define u and v , respectively, via their coordinates.

According to Pythagoras’ theorem,

(3.8)

and likewise

(3.9)

this is a more general form than Eq. (2.431), yet it relies on application of the aforementioned theorem twice. In fact,

(3.10)

abides to Eq. (2.431), as long as u xand u ydenote the projections of uonto the x ‐ and y ‐axis, respectively, and u xydenotes the projection of uonto the x 0 y plane; further application of Eq. (2.431)then supports

(3.11)

where u zdenotes the projection of uonto the z ‐axis. Insertion of Eq. (3.10)transforms Eq. (3.11)to

(3.12)

that retrieves Eq. (3.8), after taking square roots of both sides – as long as ∣ u x∣ ≡ ‖ u x‖, ∣ u y∣ ≡ ‖ u y‖, and ∣ u z∣ ≡ ‖ u z‖; a similar reasoning obviously applies to v x, v y, and v zdescribing v. The general rules of trigonometry indicate, in turn, that angle θ u(or θ v) – formed with the y ‐axis by the projection of u(or v) onto the y 0 z plane, is such that

(3.13)

and likewise

(3.14)

similar expressions can be laid out pertaining to angle ϕ u(or ϕ v) formed with the z ‐axis by the projection of u(or v) onto the x 0 z plane, viz.

(3.15)

and similarly

(3.16)

Therefore, equality of uand vrequires that ‖ u‖ = ‖ v‖ describing the same length, and θ u = θ vand ϕ u = ϕ vdescribing the same orientation – thus using up all three spatial degrees of freedom; upon inspection of the functional forms of Eqs. (3.8), (3.9), and (3.13)– (3.16), one concludes that the said three equalities unequivocally enforce u x = v x, u y = v y, and u z = v z.

The simpler operation involving vectors is addition; to be applied to uand v, the point of origin of vmust be made coincident with the point of termination of u. The vector sum, u + v , is thus the vector with the same point of origin of uand the same point of termination of v, as apparent in Fig. 3.1a. Therefore, one finds in general