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The paper is devoted to the line integral topic belonging to the section of vector calculus in Mathematical Analysis applied to the undergraduate Mechanical Engineering program. An efficient way of teaching line integrals is proposed and developed based on the elementary work/force principles. By this way, the mathematical concept of the line integral is supposed to be learned in harmony with the elementary mechanics to appreciate its diversity and to set up the right idea about the scientific area covered by Mechanical Engineering and related academic and technical fields. In terms of practical training, the research also offers two new techniques of analytical calculus for line integrals containing singularities and provides a new coherent engineering approach to deal with vector fields as integrands in line integrals. [ABSTRACT FROM AUTHOR]

The problem of equivalence between differential and integral problems is absolutely crucial when applying solution methods based on operators and their properties in function spaces. In this paper, we complement the solution of this important problem by considering the case of general derivatives and integrals of fractional order for vector functions for weak topology. Even if a Caputo differential fractional order problem has a right-hand side that is weakly continuous, the equivalence between the differential and integral forms may be affected. In this paper, we present a complete solution to this problem using fractional order Pettis integrals and suitably defined pseudo-derivatives, taking care to construct appropriate Hölder-type spaces on which the operators under study are mutually inverse. In this paper, we prove, in a number of cases, the equivalence of differential and integral problems in Hölder spaces and, by means of appropriate counter-examples, investigate cases where this property of the problems is absent. [ABSTRACT FROM AUTHOR]

The primary goal of this paper is to define Katugampola fractional integrals in multiplicative calculus. A novel method for generalizing the multiplicative fractional integrals is the Katugampola fractional integrals in multiplicative calculus. The multiplicative Hadamard fractional integrals are also novel findings of this research and may be derived from the special situations of Katugampola fractional integrals. These integrals generalize to multiplicative Riemann–Liouville fractional integrals and multiplicative Hadamard fractional integrals. Moreover, we use the Katugampola fractional integrals to prove certain new Hermite–Hadamard and trapezoidal-type inequalities for multiplicative convex functions. Additionally, it is demonstrated that several of the previously established inequalities are generalized from the newly derived inequalities. Finally, we give some computational analysis of the inequalities proved in this paper. [ABSTRACT FROM AUTHOR]

Most introductory physics courses derive the energy densities of the static electric and magnetic field for the simple cases of parallel plate capacitors and infinitely long solenoids. Authors then inform readers, usually without proof, that the energy density equations derived for the simple cases are actually the correct equations for all capacitors and inductors, regardless of their shape. The proof of the general case is typically omitted because it involves the differential calculus of vector fields. In view of this, we provide a derivation for the energy density only based on integral calculus for capacitors and inductors of any kind. The derivation, albeit seemingly complicated at first, is conceptually simple enough for introductory physics courses and does not require any knowledge of the differential calculus of vector fields. Editor's Note: The authors present a clever derivation of the static electric and magnetic field energy densities for capacitors and inductors of any shape. The derivations generalize the simple and often discussed cases of infinite parallel plate capacitors and infinitely long solenoids while requiring only a conceptual understanding of integral vector calculus. The discussion is suitable for introductory physics courses and will be of interest for teachers seeking new ways to introduce the challenging concept of field energy density. [ABSTRACT FROM AUTHOR]

The training of mathematics teachers in universities in Colombia has as a transversal axis the resolution of problems based on their social and cognitive mission pillars. In this sense, this study relates to the re-signification and construction of the concept of definite integral (DI) (integral calculus) through socio-epistemological studies, action researches, and the typology of didactic situations. The results are obtained through content analysis, didactic sequences (GeoGebra), and a discussion group. The above allows us to conclude that the validation of meanings, historical contexts, and associated social practices leads to the construction of the concept of DI as a model of mathematical analysis. This structuring of knowledge from its epistemological framework enables the exploration of mathematical objects from the basic notions that emerge in the history of humanity and didactic processes that reconstruct the evolution of the concept in society. [ABSTRACT FROM AUTHOR]

We propose a simple geometric optics analog of a gravitational lens with a refractive index equal to one at large distances and scaling like n (r) 2 = 1 + C 2 / r 2 , where C is a constant. We obtain the equation for ray trajectories from Fermat's principle of least time and the Euler equation. Our model yields a very simple ray trajectory equation. The optical rays bending, reflecting, and looping around the lens are all described by a single trigonometric function in polar coordinates. Optical rays experiencing fatal attraction are described by a hyperbolic function. We use our model to illustrate the formation of Einstein rings and multiple images. Editor's Note: This article describes a simple theoretical model for gravitational lensing. The authors analyze a graded index of refraction that reproduces the behavior for light passing near the event horizon of a black hole. The mathematical simplicity of the model permits exploration of the effects of gravitational lensing—including bending, reflection, and the formation of Einstein rings—using only integral calculus and Fermat's principle. The authors illustrate many interesting lensing phenomena with 2D and 3D graphics. The model described in this paper could be introduced as a "theoretical toy model" to complement classroom demonstrations of gravitational lensing such as a "logarithmic lens" or the stem of a wine glass, making gravitational lensing and its use in modern astrophysics accessible to introductory physics students. [ABSTRACT FROM AUTHOR]

Interval-valued maps adjust integral inequalities using different types of ordering relations, including inclusion and center-radius, both of which behave differently. Our purpose was to develop various novel bounds and refinements for weighted Hermite-Hadamard inequalities as well as their product form by employing new types of fractional integral operators under a cr-order relation. Mostly authors have used inclusion order to adjust inequalities in interval maps, but they have some flaws, specifically they lack the property of comparability between intervals. However, we show that under cr-order, it satisfies all relational properties of intervals, including reflexivity, antisymmetry, transitivity, and comparability and preserves integrals as well. Furthermore, we provide numerous interesting remarks, corollaries, and examples in order to demonstrate the accuracy of our findings. [ABSTRACT FROM AUTHOR]

In this paper, we study the singularities of Feynman integrals by compactifying the integration domain as well as the ambient space of these integrals, by embedding them in higher-dimensional space. In this compactified space, the singularities occur due to the meeting of compactified propagators at non-general position. The present analysis, which had been previously used only for the singularities of second type, is used to study other kinds of singularities viz threshold, pseudo-threshold and anomalous threshold singularities. We study various one-loop and two-loop examples and obtain their singularities. We also present observations based on results obtained, that allow us to determine whether the singularities lie on the physical sheet or not for some simple cases. Thus, this work at the frontier of our knowledge of Feynman integral calculus sheds insight into the analytic structure. [ABSTRACT FROM AUTHOR]

This work was developed to present constructive steps of multiple integrals using the open-source software Geogebra. The main focus was directed towards creating three-dimensional graphs of integrals through Riemann sums in two variables. Some practical examples are developed to demonstrate the reliability of the presented results, which are compared using traditional algebraic methods and computed in Geogebra. With this, we aim to highlight the potential of Geogebra in teaching integral calculus and make the graphical visualization process less exhaustive. [ABSTRACT FROM AUTHOR]

The correct derivation of integral inequalities on fuzzy-number-valued mappings depends on applying fractional calculus to fuzzy number analysis. The purpose of this article is to introduce a new class of convex mappings and generalize various previously published results on the fuzzy number and interval-valued mappings via fuzzy-order relations using fuzzy coordinated ỽ-convexity mappings so that the new version of the well-known Hermite–Hadamard (H-H) inequality can be presented in various variants via the fractional integral operators (Riemann–Liouville). Some new product forms of these inequalities for coordinated ỽ-convex fuzzy-number-valued mappings (coordinated ỽ-convex FNVMs) are also discussed. Additionally, we provide several fascinating non-trivial examples and exceptional cases to show that these results are accurate. [ABSTRACT FROM AUTHOR]

This paper presents a novel generic asymptotic expansion formula of expectations of multidimensional Wiener functionals through a Malliavin calculus technique. The uniform estimate of the asymptotic expansion is shown under a weaker condition on the Malliavin covariance matrix of the target Wiener functional. In particular, the method provides a tractable expansion for the expectation of an irregular functional of the solution to a multidimensional rough differential equation driven by fractional Brownian motion with Hurst index H < 1 / 2 , without using complicated fractional integral calculus for the singular kernel. In a numerical experiment, our expansion shows a much better approximation for a probability distribution function than its normal approximation, which demonstrates the validity of the proposed method. [ABSTRACT FROM AUTHOR]

This study aims to investigate the properties of fractional calculus theory (FCT) in the complex domain. We focus on the relationship between the theories of special functions (SFT) and FCT, which have seen recent advancements and have led to various successful applications in fields such as engineering, mathematics, physics, biology, and other allied disciplines. Our main contribution is the development of a special function, specifically the confluent hypergeometric function (CHF) on the complex domain. By deriving various implementations of fractional order derivatives and integral operators using this function, we present a new class of special functions combining certain cases of Mittag-Leffler and confluent hypergeometric functions. Moreover, a new numerical technique for solving linear and nonlinear multi-order fractional differential equations has been developed using the proposed class of functions and the point collocation method. Graphical results are shown to demonstrate the efficacy of this proposed technique and its applicability. [ABSTRACT FROM AUTHOR]

This study identifies the reasons for geodynamics variability of the coastal system within two cliff-shore sections of the southern Baltic Sea (SBS). The comparative analysis included distinct moraines and their foregrounds near the open sea (S1) and within the Gulf of Gdańsk (S2). Short-term trends indicate a direct link between landslide occurrence and increased cliff retreat. Long-term (total) values were obtained by developing the 4F MODEL for large-scale applications, based on the analysis of remote sensing and hydroacoustic data (to determine the extent of shore platforms), the modelling of higher-order polynomial functions describing their extent, followed by the integral calculus of the indicated functions within the open-source Desmos environment. The retreat dynamics for individual landslides (S1) was an order of magnitude higher (m/yr) than the average for the whole cliff section (0.17 ± 0.008 m/yr), which correlates well with medium- and long-term development tendencies and recession dynamics, revealed by the numerical modelling method, since approximately 8 ka b2k, years before 2000 CE (at S1 = 0.17 ± 0.020 m/yr, at S2 = 0.11 ± 0.005 m/yr). While the approach described in this paper can reveal, project, and simulate the dynamics of past and future trends within other cliffed coasts shaped in tideless conditions, it also proves stable moraine erosional responses to sea-level rise since the Mid-Holocene. [ABSTRACT FROM AUTHOR]

In this paper we study the everywhere Hölder continuity of the minima of the following class of vectorial integral funcionals: with some general conditions on the density . We make the following assumptions about the function . Let be a bounded open subset of , with , and let be a Carathéodory function, where and with . We make the following growth conditions on : there exists a constant such that for a.e. , for every and every with , and with , for a.e. , , and . Assuming that the previous growth hypothesis holds, we prove the following regularity result. If is a local minimizer of the previous functional, then for every , with . The regularity of minimizers is obtained by proving that each component stays in a suitable De Giorgi class and, from this, we conclude Hölder continuity. [ABSTRACT FROM AUTHOR]

In this work, we study the existence of solutions of nonlinear fractional coupled system of $\varphi $ -Hilfer type in the frame of Banach spaces. We improve a property of a measure of noncompactness in a suitably selected Banach space. Darbo's fixed point theorem is applied to obtain a new existence result. Finally, the validity of our result is illustrated through an example. [ABSTRACT FROM AUTHOR]

We give explicit presentations of the integral equivariant cohomology of the affine Grassmannians and flag varieties in type A, arising from their natural embeddings in the corresponding infinite (Sato) Grassmannian and flag variety. These presentations are compared with results obtained by Lam and Shimozono, for rational equivariant cohomology of the affine Grassmannian, and by Larson, for the integral cohomology of the moduli stack of vector bundles on. [ABSTRACT FROM AUTHOR]

We discuss the class of functions, which are well approximated on compacta by the geometric mean of the eigenvalues of a unital (completely) positive map into a matrix algebra or more generally a type $II_1$ factor, using the notion of a Fuglede–Kadison determinant. In two variables, the two classes are the same, but in three or more noncommuting variables, there are generally functions arising from type $II_1$ von Neumann algebras, due to the recently established failure of the Connes embedding conjecture. The question of whether or not approximability holds for scalar inputs is shown to be equivalent to a restricted form of the Connes embedding conjecture, the so-called shuffle-word-embedding conjecture. [ABSTRACT FROM AUTHOR]

In this article, we construct some examples of noncommutative projective Calabi–Yau schemes by using noncommutative Segre products and quantum weighted hypersurfaces. We also compare our constructions with commutative Calabi–Yau varieties and examples constructed in Kanazawa (2015, Journal of Pure and Applied Algebra 219, 2771–2780). In particular, we show that some of our constructions are essentially new examples of noncommutative projective Calabi–Yau schemes. [ABSTRACT FROM AUTHOR]

If X is a topological space and Y is any set, then we call a family $\mathcal {F}$ of maps from X to Y nowhere constant if for every non-empty open set U in X there is $f \in \mathcal {F}$ with $|f[U]|> 1$ , i.e., f is not constant on U. We prove the following result that improves several earlier results in the literature. If X is a topological space for which $C(X)$ , the family of all continuous maps of X to $\mathbb {R}$ , is nowhere constant and X has a $\pi $ -base consisting of connected sets then X is $\mathfrak {c}$ -resolvable. [ABSTRACT FROM AUTHOR]

The structure of Borel reducibility for equivalence relations on $\omega _1$ is determined. [ABSTRACT FROM AUTHOR]

Following Faber–Pandharipande, we use the virtual localization formula for the moduli space of stable maps to $\mathbb {P}^{1}$ to compute relations between Hodge integrals. We prove that certain generating series of these integrals are polynomials. [ABSTRACT FROM AUTHOR]

The application of fractional calculus to obtain dynamic models for supercapacitors represents an alternative approach to obtaining simpler and more accurate models. This paper presents a model for the supercapacitor in the time domain, based on the use of the fractional or non-integer order integral. This fractional model is compared with the conventional simple model, which is typically used in industrial applications. This fractional integral-based model provides satisfactory fits in relation to the number of parameters used in the model. Furthermore, an interpretation of the effect of the application of fractional integration is presented for constant current charging and discharging processes at constant current, using the Riemann–Liouville definition for the non-integer order integral. Supercapacitors are devices that exhibit non-linear behavior, with a distinct charging and discharging operation. There are several methods of dynamic analysis for the characterization of supercapacitors. The information extracted from these methods is essential for understanding the behavior of supercapacitors and, thus, ensuring that processes involving supercapacitors are as efficient as possible. This paper presents a dynamic analysis based on charge and discharge operations with constant currents. The conclusion is that the fractional model provides fairly accurate fits. [ABSTRACT FROM AUTHOR]

Copyright of Gaceta de la Real Sociedad Matematica Espanola is the property of Real Sociedad Matematica Espanola and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Local minimizers of integral functionals of the calculus of variations are analyzed under growth conditions dictated by different lower and upper bounds for the integrand. Growths of non-necessarily power type are allowed. The local boundedness of the relevant minimizers is established under a suitable balance between the lower and the upper bounds. Classical minimizers, as well as quasi-minimizers are included in our discussion. Functionals subject to so-called p , q -growth conditions are embraced as special cases and the corresponding sharp results available in the literature are recovered. [ABSTRACT FROM AUTHOR]

This article analyzes the use of mathematical models in statistics and mathematics undergraduate students at the Universidad del Valle to generate mathematical, technological and reflective discussions, when studying a model that describes some factors present in the social outburst occurred during 2021 in Colombia. Through qualitative methodology and case study, interviews conducted using Critical Mathematics Education and the socio-critical perspective of modeling were examined. The study of these factors was supported by the use of programming languages such as Python, Rstudio, Wolfram Alpha together with tools such as integral and differential calculus and differential equations. The results obtained show the different approaches to mathematical, technological and reflective discussions, which are related to the competences acquired during professional training; for each case, the particularities, applications, interpretations, evaluations and uses of mathematical models are detailed. [ABSTRACT FROM AUTHOR]

Our aim in this article is to present ideas linked to some objects of school mathematics that reveal transpositional connections pertinent to the development of epistemological reference models, linking them to the notions of mathematical objects of Differential and Integral Calculus. The theoretical framework is centered on Didactic Transposition and the Anthropological Theory of the Didactic. We show how some of the objects of school mathematics are the focus of the notions of the objects of Differential and Integral Calculus, as well as highlighting the transpositional connections between school mathematics and higher education mathematics. In these transpositive connections lie possibilities for the development of epistemological models of reference that make the teaching of the objects of Differential and Integral Calculus more comprehensible. [ABSTRACT FROM AUTHOR]

The objective of this article is to highlight the potential of Conway's theory compared to the classical concept of number with a view to contributing to the development of Epistemological Reference Models for teaching Differential and Integral Calculus. The search for a single answer to the epistemological question "What is a number?" has mobilized Mathematics epistemologists for centuries, considered essential for the foundation of this concept. John Horton Conway, an English mathematician from Princeton University, dedicated himself to researching this issue and resulted in a theory presented in the 1970s. In this article we bring elements about this theory highlighting its contributions to the evolution of the foundation of the concept of number. Conway's definition of number meets the complementarity of the intensional and extensional aspects of this concept, bringing advantages to Mathematics teaching. Scientific investigations and results of teaching practices in the field of teaching have encouraged questions about the importance of the role that the concept of real numbers has for learning Calculus and Real Analysis. Add to this question, and for Mathematics in general, and for the formation of analytical thinking, and for mathematical thinking? The reflections carried out in this article aim to raise epistemological and cognitive aspects about the classical construction of number, seeking to have an impact on current epistemology. [ABSTRACT FROM AUTHOR]

In this article, we resume discussions about the application of Differential and Integral Calculus in the Teaching of Chemical Kinetics in Teacher Training. Aspects aimed at demonstrating statements through the resolution of tasks in a course of studies and research as a strategy aimed at appropriating concepts in the domain of reaction kinetics, in the context of activities based specifically on the Anthropological Theory of the Didactic (ATD), based on epistemological models of reference and the dominant one around the object speed laws of chemical reaction. Along the way, the importance and reason for being of Integral and Differential Calculus techniques for teacher training and the formulation of didactic organizations that can serve as support for tasks in the exercise of the profession in high school were highlighted, highlighting the use of instruments of ICT, such as Excel, in the construction of these tasks. It was observed that throughout the course the students understood the steps proposed in the reference epistemological model by completing a crucial task, associated with the question that generated the course, which necessarily applies integration and differentiation techniques to obtain ordinary differential equations, associated with the technique analysis of the least squares method, for the kinetic characterization of a chemical system, a medicine in the simulation of the experimental determination of its expiration date. [ABSTRACT FROM AUTHOR]

In this theoretical essay, we intend to present task proposals and praxeological analyzes in an Epistemological Reference Model (MER) for teaching Differential and Integral Calculus, giving new meaning to the diffusion of the notion of limit of a function by definition. The aforementioned MER has as its epistemological and methodological assumption the Anthropological Theory of Didactics by Yves Chevallard and the ideas of Top Down and Bottom up attentional processing. Data production for this study occurs via praxeological analysis based on tasks extracted from textbooks and a priori analysis of the constructed MER. As a main result, it was observed through the a priori analysis of the tasks that made up the MER, that the tacit knowledge necessary to solve the tasks can be evoked by their structure, while in bottom-up processing the use of imagery resources helps in focusing paying attention to important conceptual aspects present in the proposed tasks. [ABSTRACT FROM AUTHOR]

This article aims to present a method of intervention in the classroom, specifically in the Differential and Integral Calculus course of a Mathematics Teaching degree, when working with students on the Intermediate Value Theorem. We base our approach on the studies by Bicudo (1991), Garnica (1992), and Garnica and Bicudo (1994) regarding a teaching methodology based on hermeneutic work with mathematical texts in the classroom. We describe the experience conducted and our understanding of the activity carried out. We present the questions posed by the students and the indications of their understanding of the ideas introduced in the studied theorem. [ABSTRACT FROM AUTHOR]

Differential and Integral Calculus (CDI) is an essential subject for teaching Mathematics and other sciences. Despite this importance, we observed student failure and high rates of failure or dropout, which justifies the relevance of considering epistemological aspects that make it possible to understand different phenomena in the teaching of this discipline. In this sense, we propose a study of epistemological elements of density and mass knowledge, through multivariational integrals, since integral is essential knowledge for the area of exact sciences. The objective of this study is to investigate the development and implementation of an intervention proposal, centered on study and research activities, that offers CDI students opportunities to explore the concept of the integral of one and more variables. To this end, we carried out an investigation through the creation and implementation of an intervention based on work with task-solving episodes, in order to analyze generalization movement(s) that students performed to define a multivariational defined integral from defined integrals of a variable. The results showed that the students mobilized a set of knowledge of multiple integrals, from the context of calculating mass in one, two and three dimensions. Expansive generalization was used to expand procedural issues in the calculation of an integral, while reconstructive generalization was used to understand structural aspects of the Riemann integral of more than one variable. [ABSTRACT FROM AUTHOR]

The objective of this article is at identifying epistemological obstacles manifested in learning Limit of real functions with a real variable and associate them with categories proposed by previous studies. Thereunto, we collect productions from the Theses and Dissertations Catalog (CTD in Portuguese acronym) by Capes, with filters for academic master's and doctorate productions published in the last ten years. We developed a theoretical framework based on researchers who categorize epistemological obstacles within the content of Limit, which were also a reference for the works selected in CTD. Categories proposed by the authors were associated with each other, and based on this joint discussion, a specific categorization was created for grouping epistemological obstacles discussed by these authors. This categorization supported the analysis of difficulties reported in five dissertations and one thesis, which described students' difficulties regarding various aspects involving Limit. From the analysis carried out, it was concluded that the most common difficulties are associated with obstacles categorized into E1, E2 and E4: Complexity of basic mathematical objects, Notion and formalization of Limit and Calculus Ruptures. [ABSTRACT FROM AUTHOR]

The aim of this research article is to share a study conducted within the epistemological phase of mathematical theory in the context of the sciences. This study, developed by Patricia Camarena Gallardo, involves the development, presentation, and analysis of a didactic intervention aimed at revisiting the study of functions in an introductory course on differential and integral calculus. This approach is based on a contextualized problem in civil engineering, specifically a situation related to the dynamic analysis of a frame. The development of the intervention, with an expected duration of 12 hours per class, was methodologically supported by the analysis of a textbook on structural mechanics, as well as two calculus textbooks. Additionally, the study considered the curriculaof calculus courses, topics related to the historical development of the notion of function, the epistemological obstacles encountered in this process, as well as cognitive issues related to it. [ABSTRACT FROM AUTHOR]

We consider the anisotropic integral functional of the calculus of variations ... where ci≥0 and 2 ≤pi≤pi+1 for every i = 1 ,...n-1. We exhibit a minimizer functional, for an opportune choice of the exponents pi, which turns out to be bounded everywhere and Lipschitz continuous (or even of class C¹) in an opportune subset of Ω. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study to fractional calculus for class of Ψ function. The present study is designed to study generalized fractional derivatives and find their generalized transforms called Ψ-Laplace transform and Ψ-Sumudu transform. Moreover, find the analytical solutions of some applications in physics the form of generalized fractional derivatives by transform technique. [ABSTRACT FROM AUTHOR]

This paper deals with the quasi-instantaneous determination (in a single-step response time) of apparent, active, and reactive (i.e., blind and distortion) power mean values including the total power factor, total harmonic distortion, and phase shift of fundamentals of a power electronic and electrical system (PEES) using the ip-iq method, which is the main contribution of the paper. The power components' mean values are investigated during the transient and steady states. The power components' mean values can be determined directly from phase current and voltage quantities, using an integral calculus over one period within the next calculation step and using moving average and moving rms techniques (or digital filtering). Consequently, the power factor can be evaluated with known values of a phase shift of fundamentals (using a Fourier analysis). The results of this study show how a distortion power component during transients is generated even under a harmonic supply and linear resistive–inductive load. The paper contains a theoretical base, modeling, and simulation for the three and single phases of the transients in power electronic systems. The worked-out results can be used to determine and size any PES. The presented approach brings a detailed time waveform verified by simulations in Matlab/Simulink 2022a and the Real-time HW Simulator Plecs RT Box 1. [ABSTRACT FROM AUTHOR]

In this study, a new generalized fractal–fractional (FF) derivative is proposed. By applying this definition to some elementary functions, we show its compatibility with the results of the FF derivative in the Caputo sense with the power law. The main elements of classical differential calculus are introduced in terms of this new derivative. Thus, we establish and demonstrate the basic operations with derivatives, chain rule, mean value theorems with their immediate applications and inverse function's derivative. We complete the theory of generalized FF calculus by proposing a notion of integration and presenting two important results of integral calculus: the fundamental theorem and Barrow's rule. Finally, we analytically solve interesting FF ordinary differential equations by applying our proposed definition. [ABSTRACT FROM AUTHOR]

It is the purpose of the present paper to obtain certain fixed point outcomes in the sense of C*-algebra valued metric spaces. Here, we present the definitions of the gauge function, the Bianchini-Grandolfi gauge function, a-admissibility, and (α, β)-admissible Geraghty contractive mapping in the sense of C*-algebra. Using these definitions, we define (α, β)-Bianchini-Grandolfi gauge contraction of type I and type II. Next, we prove our primary results that the function satisfying our contraction condition has to have a unique fixed point. We also explain our results using examples. Additionally, we discuss some consequent results that can be easily obtained from our primary outcomes. Finally, there is a useful application to integral calculus. [ABSTRACT FROM AUTHOR]

We study Γ-convergence of nonconvex integrals of the calculus of variations in strongly connected sets when the integrands do not have polynomial growth and can take infinite values. Applications to homogenization of unbounded integrals in strongly perforated sets are also developed. [ABSTRACT FROM AUTHOR]

In this paper, we first construct an integral identity associated with tempered fractional operators. By using this identity, we have found the error bounds for Simpson's second formula, namely Newton–Cotes quadrature formula for differentiable convex functions in the framework of tempered fractional integrals and classical calculus. Furthermore, it is also shown that the newly established inequalities are the extension of comparable inequalities inside the literature. [ABSTRACT FROM AUTHOR]

It has been shown that the findings of d -metric spaces may be deduced from S -metric spaces by considering d ϖ , ϰ = Λ ϖ , ϖ , ϰ . In this study, no such concepts that translate to the outcomes of metric spaces are considered. We establish standard fixed point theorems for integral type contractions involving rational terms in the context of complete S -metric spaces and discuss their implications. We also provide examples to illustrate the work. This paper's findings generalize and expand a number of previously published conclusions. In addition, the abstract conclusions are supported by an application of the Riemann-Liouville calculus to a fractional integral problem and a supportive numerical example. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we use differential forms to prove a number of theorems of integral vector calculus that are rarely found in textbooks. Two of them, as far as the author knows, have not been published before. Some possible applications to problems in physics are shared including a general approach for computing net forces and torques on current-carrying loops that yields insights that are not evident from the standard approach. Readers who are familiar with differential forms will enjoy seeing how they can be employed to prove several new vector calculus identities. But even readers who do not follow those derivations will benefit from seeing how these identities can be employed to find forces and torques on current-carrying loops. [ABSTRACT FROM AUTHOR]

Copyright of Revista Foco (Interdisciplinary Studies Journal) is the property of Revista Foco and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

For learning mathematics, mathematical skills are needed, some of which are mathematical reasoning skills, mathematical critical thinking skill, and mathematical literacy skills. This research aims to obtain information regarding the impact of mathematical reasoning and critical thinking skills on mathematical literacy skills. This research used a quantitative approach using an associative method with a correlational technique. The sample of this research was comprised 51 students who took integral calculus course in the Department of Mathematics and Mathematics Education, Faculty of Mathematics and Science in Universitas Negeri Jakarta, which were collected randomly using simple random sampling. The statistical analysis used in this research was multiple regression analysis. The results of this research showed that: 1) There was a positive impact of mathematical reasoning on mathematical literacy. 2) There was a positive impact of mathematical critical thinking skill on mathematical literacy. 3) There was an impact of both mathematical reasoning skill and mathematical critical thinking skill together on mathematical literacy. Further research is needed related to the impact of mathematical reasoning and critical thinking skills on mathematical literacy skills reviewed from the student's initial mathematical skill. [ABSTRACT FROM AUTHOR]