Calculus Of Several Variables Research Articles

The effect of blended learning on mathematics cognitive demand and cognitive load in learning calculus of several variables at universities in Tigray, Ethiopia

The overall objective of this study was to examine the effect of blended learning on improving students’ mathematical cognitive demand and their extraneous and germane cognitive load in learning calculus of several variables. Guided by constructivist and cognitive load theories, a quasi-experimental design was conducted in three public universities. A total of 67 second-year undergraduate mathematics students participated for 4 months. The study used a mixed research method with social constructivism augmented by pragmatism. Data were analysed using analysis of co-variance procedure and thematic analysis and discussed using both quantitative and qualitative methods. The findings revealed that the treatment group students’ mathematical cognitive demand (treatment (TG1) (M = 70.9, SD = 10.8) and treatment (TG2) (M = 69.3, SD = 20.5) were significantly higher than the posttest scores of students in the comparison group in CG (M = 58.8, SD = 13.5), (p = 0.00)) with effect size of 0.25. Moreover, their germane cognitive load had improved and extraneous cognitive load reduced significantly compared to the comparison group. To conclude, mathematics cognitive demand can be improved with blended learning. These findings imply that blending mathematics instruction in higher education institutions is useful to policymakers. The finding also revealed that cognitive load was found a negative and significant predictor of mathematics cognitive demand.

An alternative way of defining integration in multivariable calculus

In undergraduate calculus of several variables, double and triple integrals are usually defined as limits of certain Riemann sums. The existence of the integral, as well as the integration formulas, are stated without proof since they require more advanced mathematics. In this article, an alternative and straightforward way of defining multiple integrals is proposed where the usual integration formulas follow as a direct application of the definition. The underlying idea is to map the arbitrary region of integration to an n-dimensional open interval, and integration over the latter is defined via the usual iterated integral. Moreover, the substitution formula is taken as a definition. Numerous illustrative examples are provided.

How to correctly answer ‘Is the optimal critical point a local minimizer?’ in Calculus courses

This short note discusses how the optimality conditions for minimizing a multivariate function subject to equality constraints have been covered in some undergraduate Calculus courses. In particular, we will focus on the most common optimization problems in Calculus of several variables: the 2 and 3-dimensional cases. So, along with sufficient conditions for a critical point to be a local minimizer, we also present and discuss counterexamples for some statements that can be found in the literature of undergraduate Calculus related to Lagrange Multipliers, such as ‘between the critical points, the ones which have the smallest image (under the function) are minimizers’ or ‘a single critical point (which is a local minimizer) is a global minimizer’.

Computational geometry applied to develop new metrics of road and edge effects and their performance to understand the distribution of small mammals in an Atlantic forest landscape

Roads negatively affect many vertebrate species, whereas edge effect may favor some generalist species. This study aims to: 1) present a new way to calculate "line integral effects", represented by LIE and AVLIE, through new computer software, making this concept accessible to a broad audience of researchers interested in the study of Road Ecology and Tropical Forest Ecology; and, 2) test the performance of LIE and AVLIE indices, applied to road effect (LIE_road and AVLIE_road) and to edge effect (LIE_edge and AVLIE_edge), other road effect indices and forest area, using a data set on small mammal abundance in a human modified landscape in the Brazilian Atlantic Forest. Road and edge effects were represented by new metrics: Line Integral Effect (LIE) and Average Integral Effect (AVLIE), calculated using Line Integral from Differential Calculus of Several Variables through new free software developed by the second author. LIE_road and LIE_edge measure the total sum of the effect of roads (represented by lines) and edges (polygons), respectively, in relation to the forest fragment (point). AVLIE_road and AVLIE_edge measure the average of road and edge effect, respectively, in relation to the same sampling point. We used generalized linear regression models to explore the relationships between the abundance of the two groups of small mammals (forest specialists and habitat generalists) and the independent variables representing road, edge and forest effects. For forest specialists, the best model included AVLIE_road (negatively associated with abundance) and AVLIE_edge (negatively associated), while for habitat generalists, the best model included AVLIE_road (negatively associated) and LIE_edge (positively associated). Thus, there are more small mammals where road effect is lower. Forest fragments with higher edge effect showed more habitat generalists and less forest specialists. LIE and AVLIE could be useful metrics to explore edge effect separately to road effect on wildlife in forest fragments.

A new approach to the concept of linearity. Some elements for a multiplicative linear algebra

ABSTRACTAn arithmetic based on the multiplication is the base of the Bigeometric (and Multiplicative) Calculus, a type of non-Newtonian Calculus which has had uses in the digital processing of images. In this work, we present the fundamental elements to construct a proposal of Multiplicative Linear Algebra with view of being a support for a Multiplicative Calculus in Several Variables and its applications. Some important concepts such as vector space, linear transformation and matrix representation are reviewed. In particular, we give a meaning and context to the operation: matrix raised to matrix.

Multivariate generalized Gram–Charlier series in vector notations

The article derives multivariate Generalized Gram-Charlier (GGC) series that expands an unknown joint probability density function (\textit{pdf}) of a random vector in terms of the differentiations of the joint \textit{pdf} of a reference random vector. Conventionally, the higher order differentiations of a multivariate \textit{pdf} in GGC series will require multi-element array or tensor representations. But, the current article derives the GGC series in vector notations. The required higher order differentiations of a multivariate \textit{pdf} in vector notations are achieved through application of a specific Kronecker product based differentiation operator. Overall, the article uses only elementary calculus of several variables; instead Tensor calculus; to achieve the extension of an existing specific derivation for GGC series in univariate to multivariate. The derived multivariate GGC expression is more elementary as using vector notations compare to the coordinatewise tensor notations and more comprehensive as apparently more nearer to its counterpart for univariate. The same advantages are shared by the other expressions obtained in the article; such as the mutual relations between cumulants and moments of a random vector, integral form of a multivariate \textit{pdf}, integral form of the multivariate Hermite polynomials, the multivariate Gram-Charlier A (GCA) series and others.

Solving the consumer’s utility-maximization problem with CES and Cobb-Douglas utility function via mathematical inequalities

This paper presents a new, non-calculus approach to solving the consumer’s utility–maximization problem with constant elasticity of substitution (CES) utility function, as well as with Cobb-Douglas utility function in case of \(n\ge 2\) commodities. Instead of using the Lagrange multiplier method or some other method based on differential calculus of several variables which might give complicated terms and equations difficult to handle, the utility–maximization problems are solved by using Jensen’s inequality and weighted arithmetic-geometric mean (weighted AM–GM) inequality. In comparison with calculus methods, such approach does not require checking first and second order conditions.

Closed formulas for the polyharmonic operator under spherical symmetry

One of the simplest formulas of the calculus of several variables says that, when applied to spherical symmetric functions $u=u(\rho)$ in $\mathbb R^n$, the Laplace operator takes the form $\Delta u=u''(\rho)+(n-1)u'(\rho)/\rho$. In this paper, we derive the analogous explicit expression for the polyharmonic operator $\Delta^k$ in the case of spherical symmetry. Moreover, if $B$ is a ball centered at the origin and $u\in H^k_0(B)$ is spherical symmetric, then, we deduce the functional \[ J[u]= \begin{cases} \displaystyle \frac{1}{2}\int_{\Omega} (\Delta^{k/2} u(x))^2\,dx&\text{if $k$ is even}\cr \displaystyle \frac{1}{2}\int_{\Omega} |\nabla \Delta^{(k-1)/2} u(x)|^2\,dx&\text{if $k$ is odd},\cr \end{cases} \] of which $(-\Delta)^k$ is gradient as a sum of integrals in one variable. The results are based upon nontrivial combinatorial identities, which are proved by means of Zeilberger's algorithm.

DESARROLLO DE UN MODELO EPISTEMOLÓGICO DE REFERENCIA EN TORNO A LA MODELIZACIÓN FUNCIONAL

En este trabajo se presenta el diseño matemático de una pequeña parte de un modelo epistemológico de referencia que, una vez completado, deberá sustentar la organización didáctica de un proceso de estudio que incluya: el desarrollo de la modelización funcional con parámetros, la razón de ser del cálculo diferencial elemental en la última etapa de secundaria y los primeros desarrollos del cálculo en varias variables en la universidad. Aquí nos concentramos en el momento en que ya se dispone de la derivada como herramienta de trabajo para mostrar el desarrollo progresivo y la completación relativa de las praxeologías matemáticas que se construyen a medida que se avanza en los diferentes niveles de la modelización funcional. Esta propuesta se hace en el ámbito de la Teoría Antropológica de lo Didáctico y constituye una pequeña contribución al objetivo didáctico de situar la modelización matemática como un instrumento que permite articular y dar sentido a la matemática escolar.

Networked Assembly of Affine Physical System Models

Engineering design is evolving into a global strategy that distributes engineering effort to team members around the world. Because modern engineering design uses analytical models, model information must be distributed globally through computer networks. This strategy would be improved if component suppliers were able to efficiently provide dynamic models of supplied components. Furthermore, to use these component models, they must be efficiently assembled to obtain a dynamic model of a product using them. Four characteristics are needed to enable this distribution and assembly process. These characteristics are a unique standard model format, an exchange of model information through a single-query network transmission, external component models protecting proprietary internal design details, and, finally, a recursive assembly process. The modular model assembly method (MMAM) (Radcliffe et al., 2009, “Networked Assembly of Mechatronic Linear Physical System Models,” ASME J. Dyn. Syst., Meas., Control, 131, p. 021003) is a model assembly algorithm that satisfies these requirements. The MMAM algorithm assembles linear physical system models with dynamic stiffness matrices. In an affine system, deviations in the inputs and outputs exhibit a proportional relationship, but the outputs of the system are nonzero at zero input (Buck and Willcox, 1971, Calculus of Several Variables, Houghton Mifflin, Boston). One motivation for developing a process to assemble affine systems is the wide use of such models resulting from local linearization of general differentiable nonlinear physical system models about a nonzero, but constant, operating point. This paper provides the first general approach to the “operating point problem,” where the operating points of each individual component are solved as a function of the desired operating point of the model of an assembly of those components. The solution of this problem allows the assembly of linearized system models at any requested system operating point. This paper extends the MMAM to nonlinear affine system models. The MMAM uses internet agents to provide external models of components when requested by either users or other model agents. Assembly agents use the models provided by component agents to build an analytical model using models provided by component agents and assembly constraints within the assembly model agent. MMAM models are supplied in a standard form that allows an assembly agent to put together efficiently a model of the assembly that is also in the standard form. The process is recursive and facilitates hierarchical use of agents to efficiently build assemblies of assemblies to any level of complexity.

A computer laboratory program in engineering mathematics to enhance mathematical conceptualisation

This article describes a laboratory supplementary program that was integrated into a traditional mining engineering mathematics unit. The practical classes consisted of computer investigations designed to help develop mathematical concepts. The program described here was mainly web based and did not directly rely on a computer algebra system for its implementation. An evaluation of the program is included. References S. Cunningham. The visualization environment for mathematics education. In Visualization in Teaching and Learning Mathematics , ed. W. Zimmermann and S. Cunningham, 67--76. USA, Mathematical Association of America, 1991. A. Franco, P. Franco, A. Garcia, F. Garcia, F. J.Gonzalez, S. Hoya, G. Rodriguez, and A. de la Villa. Learning calculus of several variables with new technologies. The International Journal of Computer Algebra in Mathematics Education , 7 (4), 295--309, 2000. B. E. Garner and L. E. Garner. Retention of concepts and skills in traditional and reformed applied calculus. Mathematics Education Research Journal , 13 (3), 165--184, 2001 S. Habre. Visualization enhanced by technology in the learning of multivariate calculus. The International Journal of Computer Algebra in Mathematics Education , 8 (2), 115--130, 2001. B. H. Hallet. Visualization and calculus reform. In Visualization in Teaching and learning Mathematics , ed. W. Zimmermann and S. Cunningham, 121--126, 1991. USA, Mathematical Association of America F. Marton and R. Saljo. Approaches to learning. In eds. F. Marton, D. Hounsell and N. Entwistle, The Experience of Learning , 36--55, 1984. Scottish Academic Press, Edinburgh. R. Moreno and R. Mayer. Verbal redundancy in multimedia learning; When reading helps listening. Journal of Educational Psychology , 94 (1), 153--163, 2002. L. D. Murphy. Computer algebra systems in calculus reform, MSTE, University of Illinois at Urbana-Champaign, 1999. http://mste.illinois.edu/users/Murphy/Papers/CalcReformPaper.html M. Pemberton. Integrating web-based maple with a first year calculus and linear algebra course. Proceedings of the 2nd International Conference on the Teaching of Mathematics , Hersonissos, Greece, July 2002. http://www.math.uoc.gr/ ictm2/Proceedings/pap316.pdf R. Pierce and K. Stacey. Observations on students' responses to learning in a cas environment. Mathematics Education Research Journal , 13 (1), 28--46, 2001. M. D. Roblyer. Integrating Educational Technology Into Teaching (4th Ed.), 2006. Pearson, New Jersey, USA. J. Stewart. Calculus (5th Ed.), 2003. Brooks/Cole, Belmont, USA. E. J. Tonkes, B. I. Loch and A. W. Stace. An innovative learning model for computation in first year mathematics. International Journal of Mathematical Education in Science and Technology , 36 (7), 751--759, 2005. L. M. Villarreal. A step in the positive direction: Integrating a computer laboratory component into developmental algebra courses. Mathematics and Computer Education , 37 (1), 72--78, 2003. S. Vinner. The pseudo-conceptual and the pseudo-analytical thought processes in mathematics learning. Educational Studies in Mathematics , 34 (2), 97--129, 1997. P. Vlachos and A.K. Kehagias. A computer algebra system and a new approach for teaching business calculus. The International Journal of Computer Algebra in Mathematics Education . 7 (2), 87--104, 2000.

Application of Functional Calculus to Complex Cauchy Problems

In this paper, we study an approach of functional calculus in several variables developed for a particular case of commuting operators. This approach, which differs from the classical one, originates in the works of Fan-tappie. It is useful to obtain explicit solution to partial differential equations and holomorphic Cauchy problems, as well as to make a link between the holomorphic functional calculus, the theory of analytic functionals, and the Fantappie transform of analytic functionals.

Calculus problems for a new century, edited by Robert Fraga. Pp427. $25. 1993. ISBN 0-88385-084-2 (Mathematical Association of America).

Part I. Calculus: I: Derivatives: 1. Arbitrating disputes: maximizing quadratic polynomials finds a fair solution to a labor dispute 2. Fitting lines to data: minimizing quadratic polynomials finds the regression line to a data set 3. Somewhere within the rainbow: minimizing trigonometric functions explains properties of the rainbow 4. Three optimization problems in computing: maximizing functions shows how to store and transmit data most efficiently 5. Newton's method and fractal patterns: Newton's method generates fractal designs in the complex plane Part II. Calculus I: Differential Equations and Integrals: 6. How old is the Earth? The exponential decay of rubidium is used to date very old rocks 7. Falling raindrops: four differential equation models describe how a raindrop falls 8. Measuring voting power: Integrals of polynomials give a measure of voting power 9. How to tune a radio: trigonometric integrals explain tuning a radio 10. Volumes and hypervolumes: integrals calculate the volumes of objects in three and higher dimensions Part III. Calculus II: 11. Reliability and the cost of guarantees: probability integrals predict how long equipment will last 12. Queueing systems: differential equations and limits analyze waiting lines 13. Moving a planar robot arm: derivatives guide a robot arm along a curve 14. Design curves: parametric curves are used in computer-aided design of automobiles 15. Modeling the AIDS epidemic: differential equations model the spread of AIDS 16. Speedy sorting: integrals, limits, and L'Hopital's rule compare the speed of sorting algorithms Part IV. Calculus of Several Variables: 17. Hydro-turbine optimization: lagrange multipliers allocate water to hydroelectric turbines 18. Portfolio theory: lagrange multipliers choose a best stock-market portfolio.

Operational calculus in several variables

Operational calculus in several variables

An intuitive approach to calculus

“A derivative is a linear transformation which approximates a function near to a point.” I heard this definition for the first time in 1964, in a lecture at Oxford on the calculus of several variables (the lecturer was D.A. Edwards). The words came as a revelation, and from then on I had a new conception of differential calculus. At school, I thought that I knew about calculus because I could write out proofs of differentiation ‘from first principles’. I enjoyed going through the hoops and proving that δy/δ tends to a limit, but somehow the idea of differentiation never became ‘part of me’.

General power umbral calculus in several variables

In this paper we give a version of multivariate general power umbral calculus with the starting point based on the theory given by S. Roman and A. Brini. We define a topological linear space pair which has a duality more definite than that considered by the above authors.

Alexis Fontaine's route to the calculus of several variables

In the years around 1732 Alexis Fontaine des Bertins (1704–1771) member of the French Royal Academy of Sciences from 1733 on, worked out a dualoperator differential calculus to solve problems in “families of curves.” Within six years a calculus of several variables had emerged out of this work. These developments, discussed at greater length elsewhere [ Greenberg 1981, 1982 ], are summarized, with additional general reflections and the correction of one error. Together the discussions complement the excellent, recently published account of “families of curves” and the origins of partial differentiation in the works of Leibniz, the various Bernoullis, and Euler [ Engelsman 1982 ]. Common concerns motivate the works of all of these mathematicians. At the same time, certain differences in conception in Fontaine's work highlight the creativity of one of the lesser known eighteenth-century mathematicians.

The kinematical aspect of the fundamental theorem of calculus

Stokes’s theorem and Poincaré’s lemma are two well-known and remarkable aspects of the fundamental theorem of calculus in several variables. With the fanfare accorded to these two aspects, the kinematical aspect of the fundamental theorem of calculus has been unjustly neglected. This aspect, appropriately labeled the transport theorem, has special cases which are well known to workers in continuum mechanics. This paper serves to emphasize the importance of the transport theorem and to briefly survey the other aspects of the fundamental theorem of calculus.

Alexis Fontaine's integration of ordinary differential equations and the origins of the calculus of several variables

Summary Alexis Fontaine des Bertins (1704–1771) was the first French mathematician to make use of the calculus of several variables in the integration of ordinary differential equations (1738). In this paper I argue that this usage evolved from Fontaine's ‘fluxio-differential method’ of the early 1730s. In this way I extend the thesis enunciated in an earlier paper in this journal.

Alexis Fontaine's ‘Fluxio-differential method’ and the origins of the calculus of several variables

Summary Alexis Fontaine des Bertins (1704–1771) was the first French mathematician to introduce what we would now regard as results in the calculus of several variables. One example is Fontaine's theorem nF = (∂F/∂x)x + (∂F/∂y)y of 1737 for homogeneous expressions F of degree n in x and y. Many years later Fontaine indicated this particular result to have been ‘a continuation of the method of solution’ introduced by him in 1734 to solve the problem of the tautochrones. It is tempting to disregard this announcement, since the method applied to the tautochrones was a method of variations and not manifestly an exercise in the calculus of several variables. Do we have just another case of a mathematician's confusion about the origins of his earlier work? In this paper I describe Fontaine's possible intentions in his remarks.