This paper is a self-contained companion to the author's first deposit (v1, zenodo.org/records/20085431), which established an exact double integral formula for the unique zero x* of a continuous function f:[a,b]→R under minimal conditions (continuity, f(a)>0, f(b)<0, uniqueness). The v1 formula involves integration over the unbounded domain [a,b]×(0,+∞). The present work introduces the elementary change of variables u = t/(1+t), which maps (0,+∞) bijectively onto (0,1) and transforms the formula into a double integral over the compact square [a,b]×[0,1]: x* = (a+b)/2 + (1/π) ∫₀¹ ∫ₐᵇ f(x)/[(1-u)²+u²f(x)²] dx du = (a+b)/2 + (1/π) ∫ₐᵇ ∫₀¹ f(x)/[(1-u)²+u²f(x)²] du dx A single rational kernel K(x,u) = f(x)/[(1-u)²+u²f(x)²] appears on a bounded domain. We prove: (i) K ∈ L¹([a,b]×[0,1]) with exact norm π(b-a)/2(ii) Both integration orders are valid (Fubini-Tonelli)(iii) The singularity at (x*,1) is integrable and harmless(iv) The sign function sgn(f(x)) is identified as the inner integral of K in u — a consequence, not an axiom The formula is validated on f(x) = -x + cos(x) on [0,π/2], whose unique zero is the Dottie number x*≈0.739085133215161. All proofs are elementary and self-contained. No knowledge beyond standard real analysis is required.
In this paper, we generalize the work of P.T.Landsberg\cite{web1,web2} and S.S.Sidhu\cite{web3} by providing an inequality that has its main motivation from the laws of thermodynamics, in the form of a theorem which is quite useful in generating different inequalities such as the weighted AM-GM-HM inequality, the p-th power inequality , Jensen's inequality and many other inequalities.In this paper, we have not only given the thermodynamic motivation behind the inequality but we have given the required mathematical justification in the form of a straightforward rigorous proof using basic real analysis , which was not present in the works of Landsberg and Sidhu. In fact, the first statement of the theorem mathematically proves the uniqueness of the equilibrium temperature that is attained when n different bodies at different temperatures are brought in contact. The second statement of the theorem gives a mathematical proof of the fact that the process in which n bodies at different temperatures when brought in contact equilibriate to a common temperature is spontaneous,i.e., entropically favourable. Thus, this article motivates the students to come up with different mathematical results by observing the phenomena already existing in nature and also helps them to appreciate the conventional inequalities taught to them at the secondary school and undergraduate level by associating relevant physical phenomena with those inequalities.
One of the features of the text is an elaborate code which is used to refer to certain axioms, definitions, and theorems.For example, TIr is the Theorem on Irrational Numbers, which runs as follows: "If a non-zero rational number 'r is combined with an irrational number p by any one of the four operations of arithmetic, the result produced is an irrational number; in symbols, r + p, r -p, p -r, rp, r/p, p/r are irrational numbers."According to the author 1 s preface, "Experience in classroom teaching shows that the students use the code with alacrity and effectiveness in making full and concise proofs, " This reviewer feels that the book under review is a worthy addition to the literature; but on the whole he found the exposition somewhat clumsy.In a few places terms are used before they are explained (e.g."empty set," page 99) and in c some places no explanation is offered where one is clearly required, (e.g.01 is used, but never defined.Since 31 is defined, the reviewer presumes that no knowledge of factorials is assumed.)Functions are never mentioned, even though the use of functions could have simplified the treatment considerably.These objections, however, may possibly be regarded as minor.Finally, the exercises in the book are many in number and generally non-computational in nature.