Theorem Of Differential Calculus Research Articles

Novi pristup optimizaciji brzina fluida kod višecevnih razmenjivača toplote sa aspekta troškova

This paper deals with the optimum velocity of a fluid through the pipes and shell for the characteristic groups of the multiple heat exchanger in the presence of sections, starting from the total cost function, containing fixed costs, energy costs, or the force for transporting the fluid. Components optimization functions are expressed in suitable units, which enabled the establishment of a mathematical model which involves both the fluid speeds, which are optimized. For this purpose were used the mathematical transformation of thermodynamic and hydraulic relations. The optimum value of the fluid velocity is obtained by minimization of the complex functions total costs, wherein the used theorems of differential calculus that are compared with two variables function. The resulting function, as the two-dimensional problem is examined in detail by the method of mathematical analysis. During this process is taken into account and the limitations of certain parameters. By partial differentiating the optimization function, obtained a complex system of equations that can be solved by numerical mathematical method with the support of a computer program, with respect to optimized parameters which cannot be expressed explicitly. In order to efficiently obtain solutions, in addition to the analytical, was used and graphic procedure. This techno-economic approach to the problem set due to imposed additional conditions, require the iterative determination of the parameters necessary for the design. The reason for this is because had to be satisfied greater number of equations with respect to certain restrictions.

Graphic constructions of characteristic diagrams in chemical engineering and the application of differential geometry

Starting from the experimental concentration-time ( cA,t diagram this work gives the construction of the rate of reaction-time (rA,t diagram using the pure graphic method. The diagram was constructed based on the constructed tangents in arbitrary points of the starting diagram by drawing lines parallel to them in the predetermined pole. The evidence of the construction was derived using differential geometry, i.e. the main theorem of differential calculus. Differential properties between the observed values were used in the method. Starting from the analytic relations rA = rA(t) and cA = cA(t), which can be very complex (polynomes of the n-th order), and, eliminating time t in order to give a full description of the process, we obtain the analytical relation rA = rA(cA), which is then graphically represented. Hoewever, this elimination of time can also be done graphically, in a relatively simple way. After that, through the use of the integral calculus, it was shown that concentration increase in a time interval is proportional to the (rA,t) diagram surface area. Using a similar procedure, further in the paper, it was shown that the time increase is proportional to the (1/rA, cA) diagram surface area. In order for the method to be applicable in practice, we have derived relations for appropriate coefficients of proportionality. Verification of the method is illustrated by the two characteristic examples from chemical kinetics at different monotonies of the starting experimental functions.

A theoretical result on dispersion in a photonic crystal

A theoretical result on the type of dispersion of a photonic crystal is derived by using the fact that the refractive index of the crystal is a periodic function of wavelength whose period is the atomic lattice spacing of the crystal as well as by using Rolle's theorem of differential calculus.

The role of race track principles

The paper shows an alternative way of presenting differential calculus. It is shown that the Race Track Principles (RTP) (or any of the variants) is, in fact, equivalent to the Mean Value Theorem. Moreover, it is demonstrated how major theorems of differential calculus can be derived using RTP. The benefits of using RTP as a means to introduce theoretical part of differential calculus are analysed.

Discovering the cosine law via calculus

Heuristic arguments are given to explain the presence of the term ‐‐ 2bc cos a in the cosine law. The arguments presuppose the sine law and Pythagoras's theorem. The explanation hinges on the mean value theorem of differential calculus, since the triangle in question is viewed as resulting from a perturbation of a suitable right triangle.

A convenient setting for differential calculus

Cartesian closed categories play an important r6le in many aspects of mathematics. They appear in algebraic geometry, in logic, in topology. Quite naturally one is tempted to use this notion also for differential calculus. First attempts in this direction were undertaken by A. Bastiani [l] and then by A. Frolicher and W. Bucher [4]. They all used the notion of limit spaces for their generalizations of differential calculus. In retrospect one may say that A. Bastiani used a “good” definition of differentiability but the category she chose did not allow Cartesian closedness. A. Frolicher and W. Bucher took Cartesian closedness exactly as their goal. But because of their “bad” definition of continuous differentiability, their special types of limit vector spaces became increasingly complicated. Because questions of continuity and differentiability are local and topological spaces have by definition a local structure, it has many advantages to establish differential calculus in a “pure” topological setting. And then the question is: Can a differential calculus be so established in a topological setting as to obtain Cartesian closedness in the infinitely often differentiable case? For a long time it even seemed impossible to obtain anything like Cartesian closedness for continuous maps not to speak of differentiable ones. But in 1963 Gabriel and Zisman proved [6] that the category V2? of compactly generated hausdorff spaces is Cartesian closed and this is a full subcategory of the category of topological spaces. Starting from this category ‘%%, we are quite naturally led to investigate the category %%V of compactly generated vector spaces. Observing that the HahnBanach theorem is the tool for the proof of the so-called mean value theorem of differential calculus, we see that not all compactly generated vector spaces are convenient for a differential calculus. But there is a nice full subcategory of VYV and the objects of this subcategory will provide our convenient setting.

Application of neustadt's theory of extremals to an optimal control problem with a functional differential equation and a functional inequality constraint

Application of neustadt's theory of extremals to an optimal control problem with a functional differential equation and a functional inequality constraint