Differential Calculus Research Articles

INVESTIGATING SOCIAL SCIENCE STUDENTS’ UNDERSTANDING OF LIMITS THROUGH THE LENS OF THE PROCEPT THEORY

The idea of limit is central to both differential and integral calculus. It is also applicable in other disciplines such as physics, engineering, economics, etc. Because of this, conducting a study to further improve teachers’ knowledge about how social science students (whose major is economics) understand limits is of utmost importance. The reported study sought to find out how students understand the idea of limit with regard to the use of its symbolism. Sixty first year university students in the social sciences acted as the sample of the study. An adapted procept theory was used to analyse data obtained from these students through their solution to tasks on limit and explanations on their thinking and solution processes. Qualitative analysis of data indicated that some students understood the limit symbolism to be a procept while others did not. When solving the mathematical tasks, students’ difficulties emanated from: (i) their inability to coordinate the two processes, and , or and (ii) the proper use of the limit operator, and (iii) inability to realise that the simplification has led to the same response as they could not see the relationship between the results. This resulted in misalignment between their reasoning and their choice of answers where justification was required. The results also show that limits at infinity were more problematic than those of the form as where a is a constant. Students’ choice of method used depended mostly on how much efficient the method was in terms of saving time and not really on promoting understanding. The lesson learnt from the study is that when using the adjusted procept theory, the yes or no answers do not qualify to be used in concluding the level of thinking at which students are at. It is recommended that students be asked to show their working and also explain their answers so that the type of understanding that leads to their choices come to fore

RELATIONSHIP BETWEEN STUDENT DIFFERENTIAL AND INTEGRAL CALCULUS LEARNING OUTCOMES TO STUDENT DIFFERENTIAL EQUATIONS LEARNING OUTCOMES

Differential calculus and integral calculus courses are the foundations of mathematics that are important in understanding and applying differential equations. Students can take the Differential Equation course on condition that they have graduated from the Differential Calculus and Integral Calculus courses. This study aims to determine the relationship between the understanding of differential and integral calculus with students' learning outcomes of differential equations. The population in this study were all 2020 Pattimura University Teaching and Teacher Training Faculty Mathematics Education students who had taken differential calculus, integral calculus, and differential equations courses with a total sample of 33 people. The research method used is quantitative research with correlational methods. This study uses multiple regression analysis which has the aim of finding the form and influence between the independent variables and the dependent variable. Data analysis technique using multiple regression test. Based on the results of the data analysis, it was concluded that there was a significant contribution between learning outcomes of differential calculus on learning outcomes of differential equations seen in the relative contribution of learning outcomes of differential calculus on learning outcomes of differential equations based on the calculation results obtained 56.45% while the relative contribution of learning outcomes of integral calculus 43 .54% of the learning outcomes of differential equations. So it shows that there is a big influence between learning outcomes of differential calculus and integral calculus on learning outcomes of differential equation.

Differential Calculuses on Group Algebras

Differential Calculuses on Group Algebras

ВИКОРИСТАННЯ МОБІЛЬНОГО ЗАСТОСУНКУ MAXIMA У ПРОЦЕСІ НАВЧАННЯ ВИЩОЇ МАТЕМАТИКИ

Today, the educational process of higher educational institutions of Ukraine is undergoing changes and needs innovations. Recently, due to the transition to distance learning, the introduction of martial law, aerial alarms and unstable Internet connection, teachers and students must learn to quickly adapt to the relevant situations and continue to carry out online educational activities at a high level. With the transition to distance learning, the implementation of mobile learning has become relevant, which involves the use of mobile and portable devices (phones, laptops, tablets) for teaching and learning. Among all of them, it is worth noting mobile phones, because modern smartphones satisfy the basic needs for carrying out educational activities. The article examines mobile applications in the process of learning the discipline «Higher Mathematics». The following applications were selected and analyzed: Maxima, MatrixCalculator, IntroStat, Maple Calculator, Mathematics, PhotoMath. The indicators for the analysis were the ability to work offline, free of charge and support for the Android system. Applications were analyzed for solving problems from the discipline «Higher Mathematics»: calculation of matrices and determinants, statistical processing, construction of graphs of functions, differential and integral calculus, calculation of limits of numerical sequences and solving equations. It was established that the Maxima and Mathematics applications correspond to absolutely all indicators. Among all mobile applications, the Maxima application was chosen. With the help of Maxima, students do not automatically fill in numerical values to solve the problem, but use special operations to enter data. This guarantees awareness and independent solving of problems, rather than mechanical writing off ready-made calculations. The article demonstrates the practical use of the corresponding application in solving problems from the discipline «Higher Mathematics». Key words: educational process, higher educational institutions, mobile learning, mobile applications, higher mathematics.

Theoretical justification for reducing the hydraulic resistance in the confuser of the suction pipeline

The relevance of the research lies in the theoretical substantiation of the application of an improved bell (confuser) on the suction line of the pump unit of the meliorative pumping station in order to reduce hydraulic resistance compared with a cylindrical tube. The scheme of the steady nonuniform axisymmetric movement of water in a conical circular confuser is presented. The object of the research is the suction line of the pumping unit. Research Methods. The basic principles of incompressible fluid hydraulics, differential and integral calculus have been used while carrying out the research. Research results. The differential equation of the steady state nonuniform axisymmetric movement of water in the circular confusor has been calculated. As a result of its integration it has been proved that installation of the branch pipe at the head of the suction line allows one to reduce hydraulic resistance of the suction line and to increase the suction lift of the pump unit.

Basic theorems of differential calculus and their application

Basic theorems of differential calculus and their application

Thermodynamical Material Networks for Modeling, Planning, and Control of Circular Material Flows

ABSTRACT Material flow analysis (MFA) is the main methodology to assess material flow circularity. It is essentially a data-analysis-based approach whose physical foundations consist of conservation of mass. To improve both the accuracy and the repeatability of MFA models, in this paper we leverage compartmental dynamical thermodynamics merged with graph theory and control theory. The key idea is that the thermodynamic compartments and their connections can be added, removed or modified as needed to achieve a circular material flow. Thus, our methodology consists of designing thermodynamical material networks (TMNs). We also provide a physics-based definition of circularity and implement a nonlinear compartmental control, which has been possible since TMNs are highly dynamic models based on differential calculus (i.e. ordinary differential equations) rather than on arithmetic as is typical for MFA models. As we envision scalable and repeatable designs of TMNs, we made publicly available the paper source code. 1

Impact of the Flipped Classroom on Civil Engineering Students' Learning of Differential Calculus

Impact of the Flipped Classroom on Civil Engineering Students' Learning of Differential Calculus

ПРИМЕНЕНИЕ ПРОИЗВОДНОЙ ФУНКЦИИ ПРИ РЕШЕНИИ ОПТИМИЗАЦИОННЫХ ЗАДАЧ

The paper considers the application of methods of differential calculus in solving optimization problems

Una Introducción al Cálculo Diferencial e Integral Espinorial a partir de la q- Algebra Lorentziana.

In this paper we introduce the differential and integral calculus from q - Lorentzian algebra. We establish the q -spinorial variables , the q - differential calculusand the q - complex spinorial integrals. We also define the q - spinor differentialequation and the q - Lorentzian spinor differential equation. Some comments arepresented at the end of the paper.

Compatible finite element methods for geophysical fluid dynamics

This article surveys research on the application of compatible finite element methods to large-scale atmosphere and ocean simulation. Compatible finite element methods extend Arakawa’s C-grid finite difference scheme to the finite element world. They are constructed from a discrete de Rham complex, which is a sequence of finite element spaces linked by the operators of differential calculus. The use of discrete de Rham complexes to solve partial differential equations is well established, but in this article we focus on the specifics of dynamical cores for simulating weather, oceans and climate. The most important consequence of the discrete de Rham complex is the Hodge–Helmholtz decomposition, which has been used to exclude the possibility of several types of spurious oscillations from linear equations of geophysical flow. This means that compatible finite element spaces provide a useful framework for building dynamical cores. In this article we introduce the main concepts of compatible finite element spaces, and discuss their wave propagation properties. We survey some methods for discretizing the transport terms that arise in dynamical core equation systems, and provide some example discretizations, briefly discussing their iterative solution. Then we focus on the recent use of compatible finite element spaces in designing structure preserving methods, surveying variational discretizations, Poisson bracket discretizations and consistent vorticity transport.

Lifting Bratteli diagrams between Krajewski diagrams: Spectral triples, spectral actions, and AF algebras

In this paper, we present a framework to construct sequences of spectral triples on top of an inductive sequence defining an $AF$-algebra. One aim of this paper is to lift arrows of a Bratteli diagram to arrows between Krajewski diagrams. The spectral actions defining Non-commutative Gauge Field Theories associated to two spectral triples related by these arrows are compared (tensored by a commutative spectral triple to put us in the context of Almost Commutative manifolds). This paper is a follow up of a previous one in which this program was defined and physically illustrated in the framework of the derivation-based differential calculus, but the present paper focuses more on the mathematical structure without trying to study the physical implications.

Exploring mathematical connections in the context of proof and mathematical argumentation: A new proposal of networking of theories

Extended theory of mathematical connections (ETC) and theory of mathematical argumentation (TMA) based on Toulmin’s (1984) model were articulated for the study of mathematical connections activated in the argumentation process. For this purpose, a “networking of theories” was made to obtain the complementarities between both theories. Then, a class episode was selected that dealt with the demonstration of the continuity theorem of functions of real variable “<i>if a function is derivable at a point then it is continuous at that point</i>”, made by an in-service mathematics teacher of differential calculus, who participated in a non-participant observation, where his classes were videotaped. The arguments of this episode were analyzed through with Toulmin’s (1984) model, after with thematic analysis method to identify mathematical connections, and, finally, the connections in the proof and mathematical argumentation were analyzed. The main result of the research reveals that the mathematical connections play a fundamental role in the argumentation process of the episode, given that, connection is important for the establishment and identification the argument and the warrant that supports it. In addition, complementarities were found between both theories, which makes this networking a useful tool for a better analysis of mathematical argumentation processes.

Study of barley yellow dwarf virus: dynamics of aphid-plant interaction

Physics continues to be an inexhaustible source of problems that have inspired the theoretical development of mathematics, particularly in the development of the theory of dynamical systems that have their origin in the physics of the 15th century, with the beginning of differential calculus by Newton and Leibniz; differential equations are an appropriate tool for research in different areas of knowledge. Our objective is to show an example of how we can use dynamical systems in a problem framed in forestry and agricultural systems, ecology, and epidemiology. This work is inspired by one of the essential primary crops in the energy intake of the human diet, wheat; as it turns out, the productivity of this food can be affected by several biotic stresses, including viral diseases and their associated vectors. Among the viral infections is the barley yellow dwarf virus, which causes 11% to 33% yield losses in wheat fields; the virus moves through the agricultural landscape via different aphid species (vectors). The main problem is that aphids prefer colonizing plants with the virus, causing rapid growth of infected plants and substantially decreasing crop productivity; therefore, we propose to study the dynamics of aphid-plant interaction, in which the impact of aphid preference for infected plants is determined, and we obtained as a result that a key parameter is the quantity and quality of food consumed by aphids.

Investigating algebraic topological method for solving 3D Laplace equation in electrostatic boundary value problems

In this paper we investigate the adaptability of a non-conventional numerical technique called Algebraic Topological Method (ATM) for solving the Laplace equation in Electrostatic Boundary Value Problems (ESBVP). The main advantage of the ATM is that we can solve ESBVP directly in 3D domain without using complex differential or integral vector calculus tools. As a case study, we solved the problem of an ideal Parallel Disc Capacitor (PDC) using ATM by applying the Dirichlet boundary conditions at the domain boundaries. The ATM solution in the 3D domain between the disc's electrodes of PDC is presented as plots of electric potential distribution across different planes. The accuracy of scattered numerical solutions of ATM was compared with the analytical solutions for the ideal PDC. Further, the mesh convergence of ATM scattered solutions was studied by reducing the mesh size with a minimum and maximum edge length ratio as the control parameter. The ATM solution obtained from the optimum mesh size was interpolated, and the continuous electric potential distributions on various planes across the solution domain were plotted. In addition to that, the fringe field effect in a real capacitor was modeled using the ATM, and the electric potential distribution was compared with the solutions from commercial FEM software (COMSOL Multiphysics). Finally, the accuracy of ATM in case of asymmetric electrode geometries was compared with the COMSOL Multiphysics by solving the 3D Laplace equation for Semi Circular Disc Capacitor. The accuracy and solution time of the ATM for solving 3D Laplace equation in the capacitor models show that it is an efficient numerical method for solving the ESBVP.

Consistency results for the dual-wind discontinuous Galerkin method

This paper further analyzes the dual-wind discontinuous Galerkin (DWDG) method for approximating Poisson’s problem by directly examining the relationship between the Laplacian and the underlying discrete Laplacian. DWDG methods are derived from the DG differential calculus framework that defines discrete differential operators to replace continuous differential operators. We establish a priori error estimates for the DWDG approximation of Δ. Since the DWDG method does not satisfy a Galerkin orthogonality condition, we also explore the relationship between the DWDG approximation and the Ritz projection defined to satisfy an exact Galerkin orthogonality condition linked to the DWDG method. Numerical experiments are provided to validate the theoretical results.

A Novel Proof of the Desnanot-Jacobi Determinant Identity

Summary We present an alternate proof for the Desnanot-Jacobi identity using induction, basic matrix algebra and elementary differential calculus. It is an instructive demonstration of how it is occasionally necessary to utilize results from different fields of mathematics to solve a specific problem.

Fractional Hamilton’s Canonical Equations and Poisson Theorem of Mechanical Systems with Fractional Factor

Because of the nonlocal and nonsingular properties of fractional derivatives, they are more suitable for modelling complex processes than integer derivatives. In this paper, we use a fractional factor to investigate the fractional Hamilton’s canonical equations and fractional Poisson theorem of mechanical systems. Firstly, a fractional derivative and fractional integral with a fractional factor are presented, and a multivariable differential calculus with fractional factor is given. Secondly, the Hamilton’s canonical equations with fractional derivative are obtained under this new definition. Furthermore, the fractional Poisson theorem with fractional factor is presented based on the Hamilton’s canonical equations. Finally, two examples are given to show the application of the results.

Some Identities on Sums of Finite Products of the Pell, Fibonacci, and Chebyshev Polynomials

Objectives: This study will introduce some new identities for sums of finite products of the Pell, Fibonacci, and Chebyshev polynomials in terms of derivatives of Pell polynomials. Similar identities for Fibonacci and Lucas numbers will be deduced. Methods: Results are obtained by using differential calculus, combinatory computations, and elementary algebraic computations. Findings: In terms of derivatives of Pell polynomials, identities on sums of finite products of the Fibonacci numbers, Lucas numbers, Pell and Fibonacci polynomials, and Chebyshev polynomials of third and fourth kinds are obtained. Novelty: Existing research has identified identities on sums of finite products of the Fibonacci numbers, Lucas numbers, Pell and Fibonacci polynomials, and Chebyshev polynomials of the third and fourth kinds in terms of derivatives of Fibonacci polynomials or Chebyshev polynomials; identities on sums of finite products in terms of Pell polynomials, however, have not been investigated, so identities primarily in terms of Pell polynomials are obtained. Keywords: Fibonacci Numbers; Pell Numbers; Lucas Numbers; Pell Polynomials; Chebyshev Polynomials

An analysis of Irish mathematics textbook tasks in the context of curriculum change

ABSTRACT This study is concerned with the analysis of mathematical textbook tasks at the second-level in Ireland, in the context of the introduction of the revised curriculum entitled ‘Project Maths’. A total of 7635 tasks on the topics of Pattern, Sequences and Series, and Differential Calculus were analysed; these tasks were selected from editions of three textbook series available before and after the curriculum change. The analysis was carried out using a framework specially formulated for the purposes of this study, based on an amended version of the Project Maths objectives centred on its representation/characterisation of ‘synthesis and problem-solving skills’. Our findings suggest that the post-Project Maths textbook tasks offer more problem-solving opportunities compared to those in the older textbooks, but that there is still much scope for further development. Based on our analysis, it would appear that all three textbook series have neglected important objectives of the Project Maths curriculum such as providing opportunities for students to devise models/techniques and translate information from written to mathematical form.