Multivariable Calculus

In literature, the speed of light has been determined extensively through experimentation. However, only a single theoretical derivation has been offered till date. This paper aims to do the latter by using Maxwell's 3rd and 4th equations to describe orthogonal oscillating electric and magnetic fields and present an original proof for the speed of light in free space. This approach also negates the use of the partial differential wave equation, which has been used in the previous theoretical derivation.

This is a report of an Action-Process-Object-Schema Theory (APOS) based study consisting of three research cycles on student learning of the basic ideas of two-variable functions. Each of the research cycles used semi-structured interviews with students to test an initial conjecture of needed mental constructions, develop supporting classroom activities, and improve the conjecture. The article summarizes findings from each of the research cycles and shows the improvement in students' understanding of functions of two variables.

Action-Process-Object-Schema (APOS) theory and tools resulting from dialogue with the Anthropological Theory of the Didactic (ATD) were used to analyse data from semi-structured interviews and teaching materials to study students' understanding of the relationship between tangent planes and the total differential. Results of the study show students' difficulties relating these ideas and suggest a refinement of the initial genetic decomposition. They also underline aspects of the teaching materials that need to be considered to promote those constructions and development of a complete praxeology for the total differential. This study exemplifies how the dialogue between a cognitive theory and one that focuses on institutional aspects of mathematics education, can provide tools to deeply analyse the teaching and learning of a mathematical topic.

Résumé La théorie Action-Processus-Objet-Schéma (APOS), ainsi que les résultats d''un dialogue avec la Théorie Anthropologique du Didactique (ATD), ont été appliqués à l'analyse d''entretiens semi-structurés et de manuels, afin d''étudier la compréhension, par les étudiants, de la relation entre plans tangents et différentielle totale. Les résultats de l'étude montrent que les étudiants éprouvent des difficultés à relier ces idées et suggèrent des manières d'affiner la décomposition génétique initiale. Ils permettent également d''identifier des aspects des supports d'enseignement qu'il s''agit de prendre en compte pour faciliter la construction de ces notions et le développement d'une praxéologie complète pour la différentielle totale. Cette étude Int.

With the help of the National Science Foundation, the Department of Mathematics at the University of Puerto Rico in Mayaguez has developed a set of manipulatives to help students of science and engineering visualize concepts relating to points, surfaces, curves, contours, and vectors in three dimensions. This article will present the manipulatives that have been developed, describe how they have been used in multivariable calculus classes, and provide the results of initial studies on their effectiveness.

Action-Process-Object-Schema Theory (APOS) is used to study students' geometric understanding of partition of a rectangular domain and corresponding Riemann sum of an integral of a function of two variables. In this paper we mainly consider the most basic case of a partition, that consisting of a single rectangle (the domain itself). Semi-structured interviews were conducted with ten students who had just finished taking a traditional course in multivariable calculus. Results show that these students had many difficulties with even the most basic mental constructions needed to relate Riemann sum and double integral. This is an important observation since some of these mental constructions are commonly assumed to be obvious to students.

This work briefly describes, in introductory manner, some important concepts and methods in multivariate calculus, including differential operators, multivariate Taylor series, extrema, parametric curves and surfaces, and surface integration.

In this work, we generalize existing ideas of the univariate case of the time scales calculus to the bivariate case. Formal definitions of partial derivatives and iterated integrals are offered, and bivariate partial differential operators are examined. In particular, solutions of the homogeneous and nonhomogeneous heat and wave operators are found when initial distributions given are in terms of elementary functions by means of the generalized Laplace Transform for the time scale setting. Finally, the so-termed mixed time scale setting is discussed. Examples are given and solutions are provided in tabular form.

Function of several variables is one of the most important concepts in mathematics and its applications. The lack of its understanding will cause certain obstacles in the learning of next concepts or even subjects. The researchers at Universiti Teknologi Malaysia (UTM) tend to support students to overcome their deficiencies in the learning of two-variable functions by promoting mathematical thinking. The purpose of this study is to demonstrate how this method can help students in the learning of two-variable functions when they encounter ...

ABSTRACT Abstract Conceptually, the role of visual thinking is so fundamental to the understanding of calculus that it is difficult to imagine a successful calculus course which does not emphasize the visual elements of the subject. [18] With the help of the National Science Foundation (NSF-DUE 0442365), the Department of Mathematics at the University of Puerto Rico in Mayaguez has developed a set of manipulatives to help students of science and engineering visualize concepts relating to points, surfaces, curves, contours, and vectors in three dimensions. This article will present the manipulatives that have been developed, will describe how they have been used in multivariable calculus classrooms and will provide the results of initial studies on their effectiveness. Abstract Conceptually, the role of visual thinking is so fundamental to the understanding of calculus that it is difficult to imagine a successful calculus course which does not emphasize the visual elements of the subject. [18] With the help of the National Science Foundation (NSF-DUE 0442365), the Department of Mathematics at the University of Puerto Rico in Mayaguez has developed a set of manipulatives to help students of science and engineering visualize concepts relating to points, surfaces, curves, contours, and vectors in three dimensions. This article will present the manipulatives that have been developed, will describe how they have been used in multivariable calculus classrooms and will provide the results of initial studies on their effectiveness.

In multivariable calculus, the concept of multivariable function is one of the most difficult for undergraduate students to study. The main objective of this study is to establish a model of teaching and learning to support students' mathematical thinking in the learning of two-variable functions through a blended learning environment. The impact of this environment on students' learning of two-variable functions and in overcoming students' obstacles are put forward. Findings revealed that blended learning supports students' mathematical thinking ...

This article reports on an Action-Process-Object-Schema Theory (APOS) based study consisting of three research cycles on student learning of the basic idea of a two-variable functions and its graphical representation. Each of the three research cycles used semi-structured interviews with students to test a conjecture about mental constructions (genetic decomposition) students may use to understand functions of two variables, develop supporting classroom activities based on interview results, and successively improve the conjecture. The article brings together for the first time findings already reported in the literature from the first two research cycles, and the results of the third and final cycle. The final results show that students who were assigned special activities based on the research findings of the first two cycles were more likely to exhibit behavior consistent with a Process conception of function of two variables. An important contribution of the article is that it shows how different APOS research cycles may be used to successively improve students’ understanding of a mathematical notion. Also, the description of findings from the three research cycles, provides a potentially useful guide to improve student
learning of function of two variables.

Calculus as a prerequisite course to other advanced mathematics courses is one of the important and difficult courses for undergraduate students in many fields of study. Mathematical thinking is an important method to support students in the learning of calculus and specifically multivariable calculus. Researchers endeavour to support student's mathematical thinking in calculus with or without computer-based tools. The main goal of this paper is to illustrate the importance of using computer-based tools for fostering ...

We present an input-output solution for simulating the associated behavior and optimized physical needs of an environmental system. The simulations and numerical analysis determined the accurate boundary loads and areas that were required to interact for the proper physical operation of a complicated environmental system. A case study was conducted to simulate the optimum balance of an environmental system based

Abstract This article reports on an Action-Process-Object-Schema Theory (APOS) based study consisting of three research cycles on student learning of the basic idea of a two-variable functions and its graphical representation. Each of the three research cycles used semi-structured interviews with students to test a conjecture about mental constructions (genetic decomposition) students may use to understand functions of two variables, develop supporting classroom activities based on interview results, and successively improve the conjecture. The article brings together for the first time findings already reported in the literature from the first two research cycles, and the results of the third and final cycle. The final results show that students who were assigned special activities based on the research findings of the first two cycles were more likely to exhibit behavior consistent with a Process conception of function of two variables. An important contribution of the article is that it shows how different APOS research cycles may be used to successively improve students’ understanding of a mathematical notion. Also, the description of findings from the three research cycles, provides a potentially useful guide to improve student learning of function of two variables.

In the present paper, the author establish new unified integral whose integral contains products of H-function of several complex variable [1] and a general polynomials given by Srivastava [2] with general arguments. A large number of integrals involving various simpler functions follow as special cases of this integral.

Multivariable function is one of the most important concepts in the learning of advanced mathematics. We had implemented a teaching approach to support students in the learning of two-variable functions by promoting mathematical thinking in face-to-face Multivariable Calculus classroom. This study investigates the obstacles and difficulties faced by students in the learning of two-variable functions based on the mathematical thinking approach. The findings indicated that students displayed various difficulties in finding the range and sketching the graph of two-variable functions. The students’ difficulties and obstacles such as poor mastery of algebraic manipulation, poor grasp of prior knowledge or lack of it, idiosyncrasy attributed from previous mathematical experience, and restricted mental images of two-variable functions could be classified as difficulties with techniques, concepts, and studying mathematics. Based on students’ responses, the difficulties were considered mainl...