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

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Название Mathematics for Enzyme Reaction Kinetics and Reactor Performance
Автор произведения F. Xavier Malcata
Жанр Химия
Серия
Издательство Химия
Год выпуска 0
isbn 9781119490333



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      in the case of an even multiple of θ, materialized via replacement by 2n; and alternatively

      when said multiple is odd, i.e. consubstantiated in 2n + 1. Note that no need exists here to change also the form of the counting variable, because no upper limit for the summation was (deliberately) provided in Eq. (2.424) – unlike happened with Eqs. (2.388) and (2.407). For consistency between the linear expression on ι, the coefficients of ι‐dependent and ‐independent terms in both sides of Eq. (2.425) must match – so one may write

      (2.427)equation

      complemented with

      (2.428)equation

      (2.429)equation

      (2.430)equation

      2.3.3 Fundamental Theorem of Trigonometry

      which is classically known as Pythagorean equation – in honor to ancient Greek mathematician Pythagoras (570–495 BCE), historically credited for its first (recorded) proof.

Image described by caption and surrounding text.

      Despite the 400+ distinct proofs available, one may to advantage take four copies of a right triangle with sides a, b, and c – arranged inside a square with side c, as outlined in Fig. 2.11 b; the triangles share their area, ab/2 (a formula to be derived in due course), and the smaller square has side ba. The area c2 (also to be derived) of the larger square may thus be given by

      (2.433)equation

      relating length of sides in [BCD] and [ABC] opposed to right angle as left‐hand side, and relating length of sides also in [BCD] and [ABC] opposed to angle of amplitude θ (note the mutually perpendicular sides of said angles) as right‐hand side; coupled with

      upon elimination of denominators – and a similar result,