Multivariable Calculus Research Articles

The Gauss–Bonnet theorem

The Gauss–Bonnet theorem is a crowning result of surface theory that gives a fundamental connection between geometry and topology. Roughly speaking, geometry refers to the “local” properties—lengths, angles, curvature— of some fixed object, while topology seeks to identify the “global” properties that are unchanged by a continuous deformation, such as stretching or twisting. The theorem formalizes an intuitive idea: continuous changes to curvature on one region of a surface will be balanced out elsewhere, so the total curvature of the surface stays the same. Explicitly, the Gauss–Bonnet theorem says that a surface’s total curvature, defined using its local Gaussian curvature, is directly proportional to the number of holes in the surface, which comes from an invariant quantity called its Euler characteristic. The Euler characteristic is a way of classifying which surfaces can be continuously deformed into one another; as an informal example, the classic joke that “a topologist is a person who cannot tell the difference between a coffee mug and a doughnut” comes from the fact that the objects each have one hole. Even though a coffee mug and a doughnut have visibly different geometric shapes, according to the Gauss–Bonnet theorem, both objects will have the same total curvature. The proof itself is delightfully systematic: we first find the total curvature of a curve on a plane, extend that result to curves on three-dimensional surfaces, extend that result to “polygons” on surfaces, and finally the entire surface. In Section 2, we prove Hopf’s Umlaufsatz for the total curvature of a simple closed curve in R2. Sections 3, 4, and 5 introduce concepts from differential geometry to define Gaussian curvature. In Section 6, we prove the local Gauss–Bonnet theorem for the total curvature of a surface polygon. At last, in Section 7, we prove the global Gauss–Bonnet theorem for compact surfaces by covering the surface with polygons and applying the local Gauss–Bonnet theorem to each one. Our discussion focuses on exposition, and references will be given in place of tedious computations when reasonable. This paper assumes a somewhat rigorous understanding of multivariable calculus and linear algebra, as well as some elementary group theory.

Learning two-variable functions using 3D dynamic geometry

In this study, we redesigned paper-and-pencil activities and adapted them to an introductory multivariate calculus course supported by 3D dynamic geometry software. We used semi-structured interviews and students’ written productions during the semester to analyse the use of technology in supporting students’ learning. Considering the work of two students, we find that the software has transformative potential but that it did not promote students’ learning as expected. We discuss possible reasons for this.

Verification of Taylor's theorem

Multivariate function calculus is an important part of mathematical analysis courses, and most conclusions can be found and generalized in univariate calculus. However, the biggest difficulty in teaching multivariate calculus lies in its abstraction, such as Taylors theorem, multiple integral regions drawing, and integral variable transformation. At the same time, ordinary differential equations are also one of the basic courses of the profession, and dynamic systems based on ordinary differential equations have extensive applications in mathematical models of continuity problems and optimal control problems. Software such as Mathematica, Python, Matlab, etc. can solve similar problems. Therefore, this article will use the visualization and computational capabilities of Mathematica to validate important definitions and conclusions in multivariate calculus, and compare the differences among the three software in solving approximate numerical solutions of dynamic systems of ordinary differential equations from different perspectives.

Mathematical and computationalmodeling of migrations, population spread, and Location-dependent carrying capacities

In single-variable calculus, differential equation models for population dynamics are of high importance and by developing these ordinary differential equation (ODE) models, a practical and intelligible approach can be created to partial differential equations (PDEs). This paper researches focus on PDE models for migrations, population spread and location-dependent carrying capacities. However, logistic equation from which the PDE models develop include the diffusion equation, traveling wave equation, and building blocks with location-dependent parameters. This methodology is suitable for multivariable calculus lecture, evaluation task, and problem activity. The text is accompanied by interactive examples, and the article is formatted to be used as a CDF document in which some of the input can be hidden.

On Liouville’s Theorem for Conformal Maps

A theorem of Liouville asserts that the simplest angle-preserving transformations on Euclidean space—translations, dilations, reflections, and inversions—generate all angle-preserving transformations when the dimension is at least 3. This note gives a proof which uses only elementary multivariable calculus and simplifies a differential-geometric argument of Flanders.

The Triple Product Rule and the Subtleties of the Mathematical Tools Used in Thermodynamics

We have clarified several formulas from mathematics and physics. Specifically, we have shown that several standard concepts from thermodynamics, which have been recently investigated and rejected in some papers, continue to be valid. Thus, we have studied the triple product rule, the first law of thermodynamics, Mayer's relation, work, function of state, the relationship between the coefficient of thermal expansion, coefficient of isothermal compressibility and thermal coefficient of pressure. We have used multivariable calculus and properties of Jacobians and other mathematical tools. Likewise, we have provided several didactic examples concerning the interpretation of thermodynamical and mathematical formulas. We have concluded that the basement of several thermodynamical formulas is correct but full of subtleties. Thus, those formulas continue to be valid despite some claims in the recent literature.

Requirements of the undergraduate economics major: An update and comparison

The undergraduate economics curricula for most of the 793 U.S. colleges and universities that conferred an economics bachelor’s degree in 2019 are described in this article. Besides updating the prevalence of the economics major core requirements and their differences by institution type, the authors record new information on requirement variations across economics degree types, as classified by the National Center for Education Statistics, including STEM-designated degree types. They also investigate the prevalence of calculus-based intermediate courses and find that 63 percent of economics degrees require calculus for intermediate microeconomics and macroeconomics. In addition, 67 percent of degrees require single-variable calculus, 10 percent require multi-variable calculus, and 54 percent require basic econometrics (up from 41% in 2010). These requirements also vary highly by degree type.

Advancing students’ achievements in multivariable calculus education through CSCL

Advancing students’ achievements in multivariable calculus education through CSCL

Multivariable calculus textbook representation of non-cartesian coordinates: a misalignment between multivariable calculus textbook content and upper-division physics application

Upper-division undergraduate physics coursework necessitates a firm grasp on and fluid use of mathematical knowledge, including an understanding of non-cartesian (specifically polar, spherical and cylindrical) coordinates and how to use them. A limited body of research into physics students’ thinking about coordinate systems suggests that even for upper-division students, understanding of coordinate system concepts is emergent. To more fully grasp upper-division physics students’ incoming understanding of non-cartesian coordinates, the prevalence of non-cartesian content in seven popular Calculus textbooks was studied. Using content analysis techniques, a coding scheme was developed to gain insight into the presentation of coordinate system content both quantitatively and qualitatively. An initial finding was that non-cartesian basis unit vectors were absent in all but one book. A deeper analysis of three of the calculus textbooks showed that cartesian coordinates comprise an overwhelming proportion of the textbooks’ content and that qualitatively the cartesian coordinate system is presented as the default coordinate system. Quantitative and qualitative results are presented with implications for how these results might impact physics teaching and research at the middle and upper-division.

Integrating the anthropological theory of didactics in multivariate calculus education: Challenges, pedagogical shifts, and innovative activities

The anthropological theory of didactics (ATD) provides a lens to view mathematics education by placing mathematical practices within socio-cultural and historical contexts. The significance of institutions, in what regards with educational establishments, societal structures, cultural norms, and historical contexts, influences the perception and practice of mathematics in both students and teachers. This perspective is relevant for understanding the complexities of teaching multivariate calculus. Traditional approaches to this subject have been lecture-based, but with the evolution of educational paradigms, active pedagogies have emerged as vital. These include among others inquiry-based learning, collaborative learning, the flipped classroom approach, technology-enhanced active learning, and project-based learning. The challenges students face in multivariate calculus range from conceptual complexities to visualization challenges and over-reliance on memorization. In this work, we propose a holistic approach for defining class activities in multivariate calculus with emphasis in integrating visualization techniques, real-world applications, collaborative activities, and feedback mechanisms. Accordingly, four class activities, grounded in the principles of ATD and tailored to address challenges in multivariate calculus, are proposed. These activities aim to foster significant understanding, relevance, and engagement in students.

Implementing GeoGebra 3D Calculator With Augmented Reality in Multivariable Calculus Education

Implementing GeoGebra 3D Calculator With Augmented Reality in Multivariable Calculus Education

An instance of negative variance-like ‘components’ in optics by way of applying multivariable calculus to a statistical problem

In the field of optics, the design of a lens often involves analyses of the variability in an optical parameter (such as lens focal length) under manufacturing conditions, which is done in order to identify sources of high variability that may cause the optical parameter to be out of its specification. In this interdisciplinary paper, the author shares his experience in lens design, where analyses of variability are often performed through examining the partial derivatives in a multivariable Taylor series expansion of an optical parameter to second order when interactions are expected. It is shown algebraically that when the second-order expansion is applied to express a sum of squares, negative terms in the sum can occur. This instance of negative-valued “variance-like components” is interesting pedagogically in applied mathematics and statistics, as well as from an engineering standpoint, as it suggests that, based on theory, the variance of a function of random variables may not necessarily be comprised of a simple sum of positive-valued squared deviations comprising its total variance.

Analysis of Mathematical Representation Ability According to Cognitive style

<em>The aim of this research is to analyze students' mathematical representation abilities in terms of cognitive style in the multivariable calculus course taught in semester 4 of the IKIP PGRI Pontianak Mathematics Education Study Program. This research is a qualitative descriptive research in the form of case study research. Research subjects were selected based on their cognitive style which was measured using the geft instrument. Selected subjects were then asked to work on mathematical representation ability questions and then interviewed according to these questions. Based on the research results, it was found that students who had a field dependent cognitive style in symbolic representation were able to link symbols with mathematical concepts, in graphical representation they were able to create images in the form of requested graphs, while in verbal representation they were less able to convey ideas effectively orally. or written. Meanwhile, students with an independent cognitive style in symbolic representation have been able to link symbols with mathematical concepts, in graphic representation they have been able to create images in the form of requested graphics, and in verbal representation they have been able to convey ideas effectively.</em>

Applications of the Spectral Theorem: Utilizing Eigenvalues and Eigenvectors

This paper explores the applications and principles of both the real and complex spectral theorems, cornerstones of linear algebra and functional analysis. We begin with an overview of the spectral theorem and delve into the pivotal roles of eigenvalues and eigenvectors, culminating in a detailed proof of the theorem. A discerning comparison is made between the real and complex versions; notably, the latter seamlessly integrates with the fundamental theorem of algebra, an attribute absent in the former. The ensuing sections illuminate practical applications. The real spectral theorem emerges as instrumental in multivariable calculus, notably in the second derivative test, in data-driven techniques such as Principal Component Analysis (PCA), and in electrical network analyses via Kirchhoff’s matrix. The complex variant takes center stage in quantum mechanics, illuminating the Schrödinger Equation. This paper underscores the spectral theorem’s profound relevance in diverse theoretical and practical arenas.

The role of topology in the construction of students’ optimization schema for two-variable functions

This study uses Action-Process-Object-Schema theory (APOS) to examine students’ understanding of two-variable function optimization. A genetic decomposition (GD) based on the notion of Schema is proposed. This is a conjecture of mental structures and relations that students may construct to understand the optimization of these functions. The GD was tested with semi-structured interviews with eleven students who had just finished an introductory multivariable calculus course. Results show that giving explicit attention during instruction to the topological structure of the domain of the function to be optimized and the use of GD-based activities was effective in promoting students’ understanding of two-variable function optimization. On the theoretical side, the study contributes to a better understanding of the APOS notions of Schema, Schema-triad, and types of relations between Schema components that have not been used extensively in the literature and that proved to be a powerful tool to model students’ learning.

Partial Differentiation and the Coefficient of Thermal Expansion: A Multivariable Calculus Activity Connecting with Chemistry

We describe the components and implementation of an activity for multivariable calculus featuring applications to the field of chemistry. This activity focuses on the isobaric thermal expansion coefficient found using partial differentiation of the volume of an ideal gas with respect to temperature as pressure is held constant. Broader goals of this activity include enhancing calculus conceptual understanding, encouraging further interdisciplinary study and exploration, and cultivating mathematicians and scientists with broader and deeper backgrounds.

Sensitivity Analysis between Lagrange Multipliers and Consumer Budget: Utility Maximization Case

In this paper sensitivity analysis between Lagrange multipliers and total budget is discussed. The method of Lagrange multipliers is a very useful and powerful technique in multivariable calculus. In mathematical economics, utility is the vital concept that increases or decreases overall happiness of the consumers. This study tries to discuss utility maximization policy of an organization by considering two constraints: budget constraint and coupon constraint. In this article, an attempt has been taken to achieve the best result through the application of scientific method of optimization.

Dichotomy in TCR V-domain dynamics binding the opposed inclined planes of pMHC-II and pMHC-I α-helices

Ligand recognition by the human α/β T-cell antigen receptor (TCR) heterodimer protein, unlike the surface immunoglobulin (sIg) B-cell receptor, is not governed by relative binding affinity. Its interaction with the peptide (p) plus major histocompatibility complex (MHC) protein (abbrev. pMHC) likely involves some different molecular mechanism linking pMHC binding to T-cell functions. Recent analytical geometry of TCR:pMHC-II solved crystallographic structures (n = 40) revealed that each variable (V)-domain is bound in similar, yet mathematically unique orientations to its target pMHC groove. The relative position of the central cysteine of each V-domain was examined by multivariable calculus in spherical coordinates, where a simple volume element (dV) was found to describe clonotypic geometry with pMHC-II. Here, the study was expanded to include TCR:pMHC-I structures, and to model a physical mechanism, specifically involving the two directionally opposed inclined planes (IP) manifest by the two major α-helices prominent in both MHC-I and MHC-II proteins. Calculations for rotational torque of each V-domain, together with acceleration up and down the slopes of both MHC α-helices were used to estimate the time a given V-domain spends sliding down its cognate MHC IP. This V-domain rotation/sliding mechanism appears to be quantitatively unique for each TCR:pMHC V-domain (n = 40). However, there is an apparent and common dichotomy between the mobility of each V-domain with respect to the two classes of MHC proteins. Evolutionary motifs in the MHC helices support that the V-domains negotiate the opposed inclined planes of pMHC ligands in clonotypic fashion. Thus, this model is useful in understanding how mechanical forces are linked to TCR function.

Novel Techniques for Classifying Exotic Spheres in High Dimensions

Discrete calculus deals with developing the concepts and techniques of differential and integral calculus in a discrete setting, often using difference equations and discrete function spaces. This paper explores how differential-difference algebra can provide an algebraic framework for advancing discrete calculus. Differential-difference algebra studies algebraic structures equipped with both differential and difference operators. These hybrid algebraic systems unify continuous and discrete analogues of derivatives and shifts. This allows the development of general theorems and properties that cover both settings. In particular, we construct differential-difference polynomial rings and fields over discrete function spaces. We define discrete derivatives and shifts algebraically using these operators. We then study integration, summation formulas, fundamental theorems, and discrete analogues of multivariate calculus concepts from an algebraic perspective. A key benefit is being able to state unified theorems in differential-difference algebra that simultaneously yield results for both the continuous and discrete cases. This provides new tools and insights for discrete calculus using modern algebraic techniques. We also discuss applications of representing discrete calculus problems in differential-difference algebras. This allows bringing to bear algebraic methods and software tools for their solution. Specific examples are provided in areas such as numerical analysis of discrete dynamical systems defined through difference equations. The paper aims to demonstrate the capabilities of differential-difference algebra as a unifying framework for further developing the foundations and applications of discrete calculus. Broader connections to algebraic modeling of discrete physical systems are also discussed.

Endoscopic Gallbladder Drainage Conversion versus Conservative Treatment Following Percutaneous Gallbladder Drainage in High-Risk Surgical Patients.

There are no consensus guidelines for patients with acute cholecystitis undergoing percutaneous cholecystostomy who are unfit for interval cholecystectomy. The current study aimed to compare the clinical outcomes of endoscopic gallbladder drainage, i.e. conversion from percutaneous cholecystostomy (including endoscopic transpapillary gallbladder stenting and endoscopic ultrasound-guided gallbladder drainage), and conservative treatment after percutaneous cholecystostomy tube removal. This retrospective review included patients who underwent percutaneous cholecystostomy for acute cholecystitis between January 2017 and December 2020. Consecutive patients who underwent endoscopic gallbladder drainage or percutaneous cholecystostomy tube removal without interval cholecystectomy were included. Outcome measures included recurrent acute cholecystitis and unplanned readmission due to gallstone-related diseases. During the study period, 238 patients were selected (63 underwent endoscopic gallbladder drainage conversion and 175 underwent conservative treatment). Patients who underwent endoscopic gallbladder drainage conversion had lower rates of recurrent acute cholecystitis (3 [4.76%] vs 31 [17.71%], p=0.012) and unplanned readmission due to gallstone-related diseases (6 [9.52%] vs 40 [22.86%], p=0.022) than those who underwent conservative treatment following percutaneous cholecystostomy tube removal. In the univariate and multivariate analyses, calculus cholecystitis (odds ratio, 13.75; 95% confidence interval, 1.83 to 102.83; p=0.011) and conversion of endoscopic gallbladder drainage (odds ratio, 0.23; 95% confidence interval, 0.06 to 0.78; p=0.019) were significant predictive factors for recurrent acute cholecystitis. Endoscopic gallbladder drainage conversion led to more favorable outcomes than conservative treatment after percutaneous cholecystostomy tube removal. Therefore, endoscopic gallbladder drainage conversion may be considered a promising treatment option for patients undergoing percutaneous cholecystostomy who are at a high surgical risk.