Multivariable Calculus Research Articles

Mathematical Refractive Thinking Ability in Multivariable Calculus: Instrument Design and Quality of Use

<p><em>Mathematical refractive thinking ability</em><em> must </em><em>be given in every lecture because it is needed in decision making. This study aims to produce a quality mathematical refractive thinking ability instrument, especially for calculus material. Instrument quality is measured in terms of validity and reliability. Content validity, con</em><em>s</em><em>tr</em><em>u</em><em>ct validity and face validity were assessed by three material experts. The instrument was tested </em><em>by</em><em> 32 mathematics education students to measure empirical validity and reliability. This research resulted in three items of mathematical refractive thinking ability tests that are valid and have instrument reliability in the medium category.</em><em></em></p>

Serrin’s accumulation function, the First and the Second Laws of Thermodynamics

Beginning in 1979, the First and Second Laws of Classical Thermodynamics were given a new lease of life through the researches of Serrin. The definition of a positive, empirical temperature scale was shown to depend on the ideas of Maxwell and Mach. Next, a thorough description of thermodynamical systems and product systems was given, and the internal energy was defined as a function which acts as a lower potential for the difference between the heat transferred and the work done for each process. Using simple mathematics and the First Law for cycles, the existence of the internal energy function for reversible systems was derived through a well known theorem from multivariable calculus. Further, the accumulation function which defines the heat transferred between the working substance and its surroundings up to and equal to a given temperature was introduced. From this, it emerged that Clausius’ inequality for a cycle was equivalent to an integral inequality involving the accumulation function over the open, positive real numbers. It was demonstrated that this inequality leads to the definition of the absolute temperature scale. From this, entropy was defined as an upper potential to the integral appearing in the Accumulation Theorem, for each process. The existence of the entropy function for reversible processes was also proved. In addition, it was shown by Huilgol that the First Law for a cycle can be represented by an inequality, which places bounds on the work done in a cycle. Moreover, the First Law and the entropy function derived from the Second Law can be used to derive an upper bound on the work done in a cycle which is sharper than that due to Carnot. Specifically, if the cycle is irreversible, the bound includes the product of the change in entropy over the adiabatic parts of the cycle, with the least value of the absolute temperature over the heat rejecting part. In the work presented here, these ideas will be explored with a view to incorporating them, along with newer versions of the Second Law due to Serrin, in the standard curriculum of thermodynamics. In order to facilitate this inclusion, explicit forms for the accumulation function in the Otto, Diesel, Stirling and Ericsson cycles are derived.

Kemampuan Berpikir Refraktif Matematis Mahasiswa melalui Pembelajaran Peer-Assited Reflection

Refractive thinking needs to be developed in lectures because it has an essential role in decision-making. However, so far, there has not been found any relevant research that develops refractive thinking through applying a learning strategy. The purpose of this study was to obtain an overview of the enhancement of students' mathematical refractive thinking ability based on prior mathematical knowledge through the application of Peer-Assisted Reflection learning and the influence of the interaction of learning factors and the level of prior mathematical knowledge on enhancing mathematical refractive thinking ability. This research is a quantitative study with a quasi-experimental design. Using the purposive sampling technique, 73 students were selected to take the Multivariable Calculus course. The research sample was students who took the Multivariable Calculus course. The instrument used was a prior mathematical knowledge test and a mathematical refractive thinking ability test. Data analysis used t-test, t’-test, Mann-Whitney U test, Aligned Rank Transform, and two-way ANOVA. The findings of this study are: (1) The enhancement of students' mathematical refractive thinking ability who got Peer-Assisted Reflection learning is better than those who got conventional learning for all levels of prior mathematical knowledge; and (2) There is no significant effect on the interaction between learning and prior mathematical knowledge on improving mathematical refractive thinking ability.

Time series data analysis and ARIMA modeling to forecast the short-term trajectory of the acceleration of fatalities in Brazil caused by the corona virus (COVID-19).

ObjectiveThis paper incorporates the concept of acceleration to fatalities caused by the coronavirus in Brazil from time series data beginning on 17th March 2020 (the day of the first death) to 3rd February 2021 to explain the trajectory of the fatalities for the next six months using confirmed infections as the explanatory variable.MethodsAcceleration of the cases of confirmed infection and fatalities were calculated by using the concept of derivatives. Acceleration of fatality function was then determined from multivariate linear function and calculus chain rule for composite function with confirmed infections as an explanatory variable. Different ARIMA models were fitted for each acceleration of fatality function: the de-seasonalized Auto ARIMA Model, the adjusted lag model, and the auto ARIMA model with seasonality. The ARIMA models were validated. The most realistic models were selected for each function for forecasting. Finally, the short run six-month forecast was conducted on the trajectory of the acceleration of fatalities for all the selected best ARIMA models.ResultsIt was found that the best ARIMA model for the acceleration functions were the seasonalized models. All functions suggest a general decrease in fatalities and the pace at which this change occurs will eventually slow down over the next six months.ConclusionThe decreasing fatalities over the next six-month period takes into consideration the direct impact of the confirmed infections. There is an early increase in acceleration for the forecast period, which suggests an increase in daily fatalities. The acceleration eventually reduces over the six-month period which shows that fatalities will eventually decrease. This gives health officials an idea on how the fatalities will be affected in the future as the trajectory of confirmed COVID-19 infections change.

STUDENTS’ PERCEPTIONS ABOUT THE IMPLEMENTATION OF PEER-ASSISTED REFLECTION LEARNING IN MULTIVARIABLE CALCULUS LECTURES

A large class can be a problem source in learning Calculus. Peer-Assisted Reflection learning,as the solution, is a learning that combines peer analysis and self-reflection. The purpose of this study was to describe the implementation of Peer-Assisted Reflection learning based on Students’ opinions in Multivariable Calculus lecture. The research design used was pre-experimental with a one-shot case study design. The data obtained from questionnaire responses were analyzed by using descriptive statistics. The result of this study shows that thePeer-Assisted Reflection learning help Students’s carry out activities: reflecting the result of their own work, commenting on the friend’s work, and increasing understanding of the material after revising their work. On contrary, the Peer-Assisted Reflection learning has not been able to carry out the following activities: determining concepts, understanding problems in a short time, understanding the work of colleagues, and finding doubts about the correctness of the answers.

Cauchy-Weil formula, Schur-Agler type classes, new Hardy spaces of the polydisk and interpolation problems

Given n rational inner functions Φ=(Φ1,...,Φn) defining a 0-dimensional (hence finite) complete intersection in the polydisk Dn and a Weil polyhedron ΔrΦ relatively compact in Dn, we combine together the Cauchy-Weil representation formulas respectively in Dn (considered as a Weil polyhedron subordinated to the coordinate functions) and ΔrΦ in order to provide integral representations formulas involving a positive kernel provided Φ satisfies some Schur-Agler type conditions. We then study interpolation problems in H2(Dn) along the finite set Φ−1({0_}). One of the main objectives of this paper is to emphasize the important role played here by Cauchy-Weil representation formula (as in computational polynomial geometry), together with its intimate connection with multivariate residue calculus.

A Calculus Approach to Complex Variables

Summary We present a definition of holomorphic function and an approach to the Cauchy theory that presents complex function theory as a natural outgrowth of multivariable calculus.

Gamma-Ray Modified Polymer/Clay Composites: Synthesis, Characterization, and Formulation Optimization Using Multivariate Calculus and Graph Theory

To enhance the oxidation strength and crosslinking yield of ultrahigh molecular weight polyethylene (UHMWPE), its composites were prepared by mixing 1%, 2%, and 3% (by wt.) of magnesium silicate hydrous (sepiolite) during this study. These composites were irradiated with 25 kGy and 50 kGy of irradiation doses in the open air. Subsequent to irradiation, the composites were characterized for updates in structure, oxidation strength, and degree of crosslinking while estimating the values of oxidation index (OI), crosslink density (Gx), and percent crystallinity (Xc), respectively. The reaction of modified sepiolite (SP) on irradiating UHMWPE/SP composites was evident from the origination of peaks at Si-O, O-Si-O, Si-O-Si, and Mg-OH and stretching vibration at 974 cm−1, 1014 cm−1, and 1080 cm−1, respectively. In addition, the moderate negative correlation of OI (−0.46) and strong positive correlation of Gx (0.87) with the absorbed dose, made obvious from correlation analysis, confirmed the effectiveness of SP in enhancing the oxidation strength and crosslink density of UHMWPE. Furthermore, cluster analysis (CA) grouped the composites as low, moderate, and best based on dissimilarities, i.e., oxidation strength and crosslink density, which was further confirmed by principal component analysis (PCA). PCA also revealed that the OI, Gx, and absorbed dose were the active variables for this best group. After confirming the effectiveness of SP (as an active filler for enhancing the oxidation strength and crosslink density of UHMWPE) and recognizing the OI, Gx, and absorbed dose as active variables, the next step was to determine the necessary minimum amount of SP concentration and radiation dose required for the composite that outperforms all others. For this, a parametric graph theory and matrix approach was employed to rank the composites of the best group, revealing that the UHMWPE/SP composite with 1% (by wt.) of Si12O30Mg8(OH)4(H2O)4·8H2O and irradiated with 50 kGy of irradiation dose was the most suitable choice of all existing alternatives tested in this study. Although this is the first attempt, to the best of our knowledge, where a smart approach using multivariate calculus and decision-making tools was utilized for figuring the best UHMWPE composite formulation along with appropriate treatment dose, the results and methodology could be extended for any polymer of industrial scale, such as those used in medical implants, defense armor, bulletproof jackets, etc.

Generalized Conformable Mean Value Theorems with Applications to Multivariable Calculus

The conformable derivative and its properties have been recently introduced. In this research work, we propose and prove some new results on the conformable calculus. By using the definitions and results on conformable derivatives of higher order, we generalize the theorems of the mean value which follow the same argument as in the classical calculus. The value of conformable Taylor remainder is obtained through the generalized conformable theorem of the mean value. Finally, we introduce the conformable version of two interesting results of classical multivariable calculus via the conformable formula of finite increments.

Designing Interactive Videos in Online Multivariable Calculus Course to Support Student’s Critical Thinking

Emphasizing critical thinking skills in the curriculum of higher education institutions is needed to produce graduates with higher order thinking skills. Given the importance of information technology for critical thinking in learning, it is necessary to understand the key factors related to learning designs that utilize information technology in their learning environment and consider the pedagogical and information technology alignment. This study aims to describe the process of analysing, designing, and developing interactive learning videos as part of the redesign of the multivariable calculus course to support students’ critical thinking skills. This paper will describe the lessons learned from and the next steps in this research and development process.

Multibody Dynamics on Differentiable Manifolds

Abstract Topological and vector space attributes of Euclidean space are consolidated from the mathematical literature and employed to create a differentiable manifold structure for holonomic multibody kinematics and dynamics. Using vector space properties of Euclidean space and multivariable calculus, a local kinematic parameterization is presented that establishes the regular configuration space of a multibody system as a differentiable manifold. Topological properties of Euclidean space show that this manifold is naturally partitioned into disjoint, maximal, path connected, singularity free domains of kinematic and dynamic functionality. Using the manifold parameterization, the d'Alembert variational equations of multibody dynamics yield well-posed ordinary differential equations of motion on these domains, without introducing Lagrange multipliers. Solutions of the differential equations satisfy configuration, velocity, and acceleration constraint equations and the variational equations of dynamics, i.e., multibody kinematics and dynamics are embedded in these ordinary differential equations. Two examples, one planar and one spatial, are treated using the formulation presented. Solutions obtained are shown to satisfy all three forms of kinematic constraint to within specified error tolerances, using fourth-order Runge–Kutta numerical integration methods.

Estimating the Sensitivity of Fitted Parameters to Perturbations of Data with Calculus

This article provides conceptual ideas, data, and exercises, for integrating original sources of recent, state of the art, world-class life science research in the undergraduate mathematics curriculum and classroom. To this end, this article shows how one of the main goals of calculus in the life sciences, fitting parameters to data and assessing the reliability of the fitted parameters, fits in existing mathematics courses at levels from basic to multivariable calculus. The material for multivariable calculus explains methods of computation of quantitative criteria of “significance” that are currently the subject of scientific controversy. This article thus provides resources to strengthen students' mastery of calculus before, or in, mathematical modeling, probability, and statistics.

The learning and teaching of multivariable calculus: a DNR perspective

The learning and teaching of multivariable calculus: a DNR perspective

Multivariable calculus results in different countries

Multivariable calculus results in different countries

Kesalahan Pemahaman Konsep Peserta Didik dalam Menyelesaikan Soal-Soal Integral Lipat Dua pada Koordinat Polar

Penelitian ini adalah penelitian deskriptif kualitatif untuk menganalisis kesalahan pemahaman konsep peserta didik dalam menyelesaikan soal-soal integral lipat dua pada koordinat polar. Tiga puluh peserta didik yang mengambil mata kuliah kalkulus peubah banyak menjadi subjek dalam penelitian ini. Data dikumpulkan dengan menggunakan tes dan dokumentasi. Hasil jawaban tes peserta didik serta dokumentasi hasil pengerjaan peserta didik, dikoreksi, dianalisis berdasarkan indikator pemahaman konsep dan didisplay data. Analisis kesalahan jawaban peserta didik berdasarkan indikator pemahaman konsep yakni (1) menyatakan ulang konsep integral. (2) merepresentasikan konsep ke bentuk lain, (3) memilih dan menggunakan prosedur serta (4) menggunakan konsep dan algoritma pemecahan masalah. Dalam menyatakan ulang konsep integral peserta didik langsung menggunakan batas atas dan batas bawah integrasi tanpa menuliskan terlebih dahulu hasil pengintegralan. Kesalahan merepresentasikan fungsi ke grafik adalah, fungsi langsung disketsakan tanpa membuat table, memplot titik dan membuat kurva. Kesalahan dalam memilih dan menggunakan prosedur dan penggunaan algoritma pemecahan masalah dikarenakan kurangnya pengetahuan awal peserta didik mengenai materi prasyarat yakni, integral trigonometri dan fungsi trigonometri.

Analisis kemampuan pemahaman matematis pada mata kuliah kalkulus peubah banyak

This study aims to determine the ability of students’ mathematical understanding in the multi-variable calculus course. The method used in this study is a test. This test will be used to determine students’ mathematical comprehension abilities in the multivariable calculus course. The results of this study concluded that 3 students scored 50-60, 4 students scored 60-70, 6 students scored 70-80. There are several factors that cause student scores to not get perfect scores, namely in the learning process there is a need for an attractive learning model, the need for technology, etc.

Teaching multivariable calculus and tensor calculus with computer algebra software

To go from calculus of scalar functions of one variable to multivariate calculus of vector-valued functions is a steep learning curve for many students. It takes a lot of practice to get used to the new concepts such as the directional derivatives, the di.erentiability, the many types of first order di.erential operators, parameterization of surfaces and the fundamental theorems of integrals, e.g. the Divergence Theorem and the Stokes Theorem. Along the learning process of mastering the skills, the students often need to check whether the intermediate steps in the tedious calculations are correct. Unfortunately, this is beyond the capability of an ordinary calculator and the answers provided at the end of the books. This is where computer algebra software, such as Mathematica, can come to students' help. With the developed symbolic computation tools, the students can tweak a given problem, solve a new one by hand and then check the answer against the result obtained by using the computer algebra software.

Graph Calculus and the Disconnected-Boundary Schwinger-Dyson Equations of Quartic Tensor Field Theories

Tensor field theory (TFT) focuses on quantum field theory aspects of random tensor models, a quantum-gravity-motivated generalisation of random matrix models. The TFT correlation functions have been shown to be classified by graphs that describe the geometry of the boundary states, the so-called boundary graphs. These graphs can be disconnected, although the correlation functions are themselves connected. In a recent work, the Schwinger-Dyson equations for an arbitrary albeit connected boundary were obtained. Here, we introduce the multivariable graph calculus in order to derive the missing equations for all correlation functions with disconnected boundary, thus completing the Schwinger-Dyson pyramid for quartic melonic (‘pillow’-vertices) models in arbitrary rank. We first study finite group actions that are parametrised by graphs and build the graph calculus on a suitable quotient of the monoid algebra mathcal {A}[G] corresponding to a certain function space mathcal {A} and to the free monoid G in finitely many graph variables; a derivative of an element of mathcal {A}[G] with respect to a graph yields its corresponding group action on mathcal {A}. The present result and the graph calculus have three potential applications: the non-perturbative large-N limit of tensor field theories, the solvability of the theory by using methods that generalise the topological recursion to the TFT setting and the study of ‘higher dimensional maps’ via Tutte-like equations. In fact, we also offer a term-by-term comparison between Tutte equations and the present Schwinger-Dyson equations.

Can I or Can’t I?

Sometimes, we quit things because we really don't want to do them anymore. For instance, I quit piano when I was still in elementary school because I didn't enjoy it. I knew I could keep getting better if I tried, but I felt only joy on the day I knew I would never have to play another note, and I've had no remorse since. But very often, we quit things because we don't think we can do them even if we try. I quit taking math in college, because some part of me doubted my ability to succeed beyond multivariable calculus. I loved derivatives and integrals and everything I was learning at that point—and I especially loved my math professor, Robin Gottlieb. Yet the idea of progressing to upper-level courses was terrifying. Quitting is sometimes the right decision, but in the moment, when we declare, “I'm done! No more for me!” it can be very difficult to know whether we're quitting for the right reasons.

Teaching calculus with infinitesimals and differentials

Several new approaches to calculus in the U.S. have been studied recently that are grounded in infinitesimals or differentials rather than limits. These approaches seek to restore to differential notation the direct referential power it had during the first century after calculus was developed. In these approaches, a differential equation like dy = 2x·dx is a relationship between increments of x and y, making dy/dx an actual quotient rather than code language for $$\underset{h\to 0}{\mathrm{lim}}\frac{f\left(x+h\right)-f(x)}{h}$$ . An integral $${\int }_{a}^{b}2x dx$$ is a sum of pieces of the form 2x·dx, not the limit of a sequence of Riemann sums. One goal is for students to develop understandings of calculus notation that are imbued with more direct referential meaning, enabling them to better interpret and model situations by means of this notation. In this article I motivate and describe some key elements of differentials-based calculus courses, and I summarize research indicating that students in such courses develop robust quantitative meanings for notations in single- and multi-variable calculus.