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Mathematica in the classroom: new tools for exploring precalculus and differential calculus

Conceição, Ana; Pereira, José; Silva, Cátia; Simão, Cristina
Fonte: Universidade do Algarve Publicador: Universidade do Algarve
Tipo: Artigo de Revista Científica
Publicado em /04/2012 ENG
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Prémio de Melhor Artigo de Jovem Investigador atribuído pela empresa Timberlake, apresentado na 1ª Conferência Nacional sobre Computação Simbólica no Ensino e na Investigação - CSEI2012, que decorreu no IST nos dias 2 e 3 de Abril.; The main goal of this paper is to present some interactive tools, F-Tools, designed by us and implemented with the computer algebra system Mathematica, which we hope will improve the teaching-and-learning process by providing teachers and students alike with new ways to explore some of the main mathematical subjects, at the secondary and university levels, speci cally in the areas of precalculus and di erential calculus. We believe that these new tools are an important contribution to Mathematical Education, providing new ways for teaching and learning. We intend to make available several F-Tools, such as F-Linear, F-Quadratic, F-Exponential, F-Logarithm, and F-Trigonometric, at the Wolfram Demonstrations Project site.

18.013A Calculus with Applications, Fall 2001; Calculus with Applications

Kleitman, Daniel J.
Fonte: MIT - Massachusetts Institute of Technology Publicador: MIT - Massachusetts Institute of Technology
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EN-US
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Differential calculus in one and several dimensions. Java applets and spreadsheet assignments. Vector algebra in 3D, vector- valued functions, gradient, divergence and curl, Taylor series, numerical methods and applications. Given in the first half of the first term. However, those wishing credit for 18.013A only, must attend the entire semester. Prerequisites: a year of high school calculus or the equivalent, with a score of 4 or 5 on the AB, or the AB portion of the BC, Calculus test, or an equivalent score on a standard international exam, or a passing grade on the first half of the 18.01 Advanced Standing exam.

Differential calculus and integration of generalized functions over membranes

Aragona, Jorge; Fernandez, Roseli; Juriaans, Stanley O.; Oberguggenberger, Michael
Fonte: SPRINGER WIEN; WIEN Publicador: SPRINGER WIEN; WIEN
Tipo: Artigo de Revista Científica
ENG
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In this paper we continue the development of the differential calculus started in Aragona et al. (Monatsh. Math. 144: 13-29, 2005). Guided by the so-called sharp topology and the interpretation of Colombeau generalized functions as point functions on generalized point sets, we introduce the notion of membranes and extend the definition of integrals, given in Aragona et al. (Monatsh. Math. 144: 13-29, 2005), to integrals defined on membranes. We use this to prove a generalized version of the Cauchy formula and to obtain the Goursat Theorem for generalized holomorphic functions. A number of results from classical differential and integral calculus, like the inverse and implicit function theorems and Green's theorem, are transferred to the generalized setting. Further, we indicate that solution formulas for transport and wave equations with generalized initial data can be obtained as well.; FAPESP (Brazil); FAPESP-Brazil; CNPq-Brazil; CNPq (Brazil)

Differential calculus in categories. I

Molotkov, Vladimir
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 19/09/2005
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Here are considered some categorical aspects of "Differential calculus" archetype of local approximation of arbitrary morphisms by "linear" ones.; Comment: 16 pages

Bi-differential calculus and the KdV equation

Dimakis, Aristophanes; Muller-Hoissen, Folkert
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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A gauged bi-differential calculus over an associative (and not necessarily commutative) algebra A is an N-graded left A-module with two covariant derivatives acting on it which, as a consequence of certain (e.g., nonlinear differential) equations, are flat and anticommute. As a consequence, there is an iterative construction of generalized conserved currents. We associate a gauged bi-differential calculus with the Korteweg-de-Vries equation and use it to compute conserved densities of this equation.; Comment: 9 pages, LaTeX, uses amssymb.sty, XXXI Symposium on Mathematical Physics, Torun, May 1999, replaces "A notion of complete integrability in noncommutative geometry and the Korteweg-de-Vries equation"

Two-Parameter Differential Calculus on the h-Exterior Plane

Celik, Salih
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 12/12/2001
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We construct a two-parameter covariant differential calculus on the quantum $h$-exterior plane. We also give a deformation of the two-dimensional fermionic phase space.; Comment: 7 pages

Differential calculus on the h-superplane

Celik, Salih; Celik, Sultan A.; Arik, Metin
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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A non-commutative differential calculus on the $h$-superplane is presented via a contraction of the $q$-superplane. An R-matrix which satisfies both ungraded and graded Yang-Baxter equations is obtained and a new deformation of the $(1+1)$ dimensional classical phase space (the super-Heisenberg algebra) is introduced.; Comment: 14 pages

Two-parameter differential calculus on the h-superplane

Celik, Salih; Celik, Sultan A.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 13/12/2001
Relevância na Pesquisa
56.27%
We introduce a noncommutative differential calculus on the two-parameter $h$-superplane via a contraction of the (p,q)-superplane. We manifestly show that the differential calculus is covariant under $GL_{h_1,h_2}(1| 1)$ transformations. We also give a two-parameter deformation of the (1+1)-dimensional phase space algebra.; Comment: 14 pages

Logic of differential calculus and the zoo of geometric strujctures

Vinogradov, Alexandre M.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 21/11/2015
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Since the discovery of differential calculus by Newton and Leibniz and the subsequent continuous growth of its applications to physics, mechanics, geometry, etc, it was observed that partial derivatives in the study of various natural problems are (self-)organized in certain structures usually called geometric. Tensors, connections, jets, etc, are commonly known examples of them. This list of classical geometrical structures is sporadically and continuously widening. For instance, Lie algebroids and BV-bracket are popular recent additions into it. Our goal is to show that the "zoo" of all geometrical structures has a common source in the calculus of functors of differential calculus over commutative algebras, which surprisingly comes from a due mathematical formalization of observability mechanism in classical physics. We also use this occasion for some critical remarks and discussion of some perspectives.; Comment: 29 pages

Differential Calculus, Manifolds and Lie Groups over Arbitrary Infinite Fields

Bertram, Wolfgang; Glockner, Helge; Neeb, Karl-Hermann
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 24/03/2003
Relevância na Pesquisa
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We present an axiomatic approach to finite- and infinite-dimensional differential calculus over arbitrary infinite fields (and, more generally, suitable rings). The corresponding basic theory of manifolds and Lie groups is developed. Special attention is paid to the case of mappings between topological vector spaces over non-discrete topological fields, in particular ultrametric fields or the fields of real and complex numbers. In the latter case, a theory of differentiable mappings between general, not necessarily locally convex spaces is obtained, which in the locally convex case is equivalent to Keller's C^k_c-theory.; Comment: 70 pages

Conceptual differential calculus part ii: Cubic higher order calculus

Bertram, Wolfgang
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 12/10/2015
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Following the programme set out in Part I of this work, we develop a conceptual higher order differential calculus. The '' local linear algebra '' defined in Part I is generalized by '' higher order local linear algebra ''. The underlying combinatorial object of such higher algebra is the natural n-dimensional hyper-cube, and so we qualify this calculus as '' cubic ''. More precisely, we define two versions of conceptual cubic calculus: '' full '' and '' symmetric cubic ''. The theory thus initiated sheds new light on several foundational issues.; Comment: 38 pages, 3 figures (XeLatex)

Deformations of normed groupoids and differential calculus. First part

Buliga, Marius
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 06/11/2009
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Differential calculus on metric spaces is contained in the algebraic study of normed groupoids with $\delta$-structures. Algebraic study of normed groups endowed with dilatation structures is contained in the differential calculus on metric spaces. Thus all algebraic properties of the small world of normed groups with dilatation structures have equivalent formulations (of comparable complexity) in the big world of metric spaces admitting a differential calculus. Moreover these results non trivially extend beyond metric spaces, by using the language of groupoids.

Simplicial Differential Calculus, Divided Differences, and Construction of Weil Functors

Bertram, Wolfgang
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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We define a simplicial differential calculus by generalizing divided differences from the case of curves to the case of general maps, defined on general topological vector spaces, or even on modules over a topological ring K. This calculus has the advantage that the number of evaluation points growths linearly with the degree, and not exponentially as in the classical, "cubic" approach. In particular, it is better adapted to the case of positive characteristic, where it permits to define Weil functors corresponding to scalar extension from K to truncated polynomial rings K[X]/(X^{k+1}).; Comment: V2: minor changes, and chapter 3: new results included; to appear in Forum Mathematicum

Differential calculus on Hopf Group Coalgebra

Hegazi, A. S.; Morsi, W.; Mansour, M.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 25/07/2005
Relevância na Pesquisa
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In this paper we construct the Differential calculus on the Hopf Group Coalgebra introduced by Turaev [10]. We proved that the concepts introduced by S.L.Woronowicz in constructing Differential calculus on Hopf Compact Matrix Pseudogroups (Quantum Groups)[7] can be adapted to serve again in our construction.

Matched differential calculus on the quantum groups $GL_q(2,C),SL_q(2,C),C_q(2|0)$

Akulov, V. P.; Gershun, V. D.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 29/09/1995
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We proposed the construction of the differential calculus on the quantum group and its subgroup with the property of the natural reduction: the differential calculus on the quantum group $GL_q(2,C)$ has to contain the differential calculus on the quantum subgroup $SL_q(2,C)$ and quantum plane $C_q(2|0)$ (''quantum matrjoshka''). We found, that there are two differential calculi, associated to the left differential Maurer--Cartan 1-forms and to the right differential 1-forms. Matched reduction take the degeneracy between the left and right differentials. The classical limit ($q\to 1$) of the ''left'' differential calculus and of the ''right'' differential calculus is undeformed differential calculus. The condition ${\cal D}_qG=1$ gives the differential calculus on $SL_q(2,C)$, which contains the differential calculus on the quantum plane $C_q(2|0)$.; Comment: LaTeX, 22 pages

Covariant q-Differential Calculus and its Deformations at q^N=1

Kerner, R.; Niemeyer, B.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 26/04/2000
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We construct the generalized version of covariant Z_3-graded differential calculus introduced by one of us (R.K.), and then extended to the case of arbitrary Z_N grading. Here our main purpose is to establish the recurrence formulae for the N-th power of covariant q-differential D_q = d_q + A and to analyze more closely the particular case of q being an Nth primitive root of unity. The generalized notions of connection and curvature are introduced and several examples of realization are displayed for N=3 and N=4. Finally we briefly discuss the idea of infinitesimal deformations of the parameter q in the complex plane.; Comment: 16 pages, no figures

Exterior differential calculus in generalized Lie algebras/algebroids category with applications to interior and exterior algebraic/differential systems

Arcus, C. M.; Peyghan, E.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 11/12/2014
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A new category of Lie algebras, called generalized Lie algebras, is presented such that classical Lie algebras and Lie-Rinehart algebras are objects of this new category. A new philosophy over generalized Lie alge- broids theory is presented using the notion of generalized Lie algebra and examples of objects of the category of generalized Lie algebroids are pre- sented. An exterior differential calculus on generalized Lie algebras is pre- sented and a theorem of Maurer-Cartan type is obtained. Supposing that any submodule/vector subbundle of a generalized Lie algebra/algebroid is an interior algebraic/differential system (IAS/IDS) for that generalized Lie algebra/algebroid, then the involutivity of the IAS/IDS in a result of Frobenius type is characterized. Introducing the notion of exterior algebraic/differential system of a generalized Lie algebra/algebroid, the involutivity of an IAS/IDS is characterized in a result of Cartan type. Fi- nally, new directions by research in algebraic/differential simplectic spaces theory are presented.; Comment: 35 pages. arXiv admin note: text overlap with arXiv:1311.1147

The consistent reduction of the differential calculus on the quantum group $GL_{q}(2,C)$ to the differential calculi on its subgroups and $\sigma$-models on the quantum group manifolds $SL_{q}(2,R)$, $SL_{q}(2,R)/U_{h}(1)$, $C{q}(2|0)$ and infinitesimal transformations

Gershun, V. D.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 24/11/1997
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Explicit construction of the second order left differential calculi on the quantum group and its subgroups are obtained with the property of the natural reduction: the differential calculus on the quantum group $GL_q(2,C)$ has to contain the 3-dimensional differential calculi on the quantum subgroup $SL_q(2,C)$, the differential calculi on the Borel subgroups $B_{L}^{(2)}(C)$, $B_{U}^{(2)}(C)$ of the lower and of the upper triangular matrices, on the quantum subgroups $U_{q}(2)$, $SU_{q}(2)$, $Sp_{q}(2,C)$, $Sp_{q}(2)$, $T_{q}(2,C)$, $B_{L}(C)$, $B_{U}(C)$, $U_{q}(1)$, $Z_{-}^{(2)}(C)$, $Z_{+}^{(2)}(C)$ and on the their real forms. The classical limit ($q\to 1$) of the left differential calculus is the nondeformed differential calculus. The differential calculi on the Borel subgroups $B_{L}(C)$, $B_{U}(C)$ of the $SL_{q}(2,C)$ coincide with two solutions of Wess-Zumino differential calculus on the quantum plane $C_q(2|0)$. The spontaneous breaking symmetry in the WZNW model with $SL_{q}(2,R)$ quantum group symmetry over two-dimensional nondeformed Minkovski space and in the $\sigma$-models with ${SL_{q}(2,R)/U_{p}(1)}$, $C_{q}(2|0)$ quantum group symmetry is considered. The Lagrangian formalism over the quantum group manifolds is discussed. The variational calculus on the $SL_{q}(2...

Discontinuous Galerkin finite element differential calculus and applications to numerical solutions of linear and nonlinear partial differential equations

Feng, Xiaobing; Lewis, Thomas; Neilan, Michael
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
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This paper develops a discontinuous Galerkin (DG) finite element differential calculus theory for approximating weak derivatives of Sobolev functions and piecewise Sobolev functions. By introducing numerical one-sided derivatives as building blocks, various first and second order numericaloperators such as the gradient, divergence, Hessian, and Laplacian operator are defined, and their corresponding calculus rules are established. Among the calculus rules are product and chain rules, integration by parts formulas and the divergence theorem. Approximation properties and the relationship between the proposed DG finite element numerical derivatives and some well-known finite difference numerical derivative formulas on Cartesian grids are also established. Efficient implementation of the DG finite element numerical differential operators is also proposed. Besides independent interest in numerical differentiation, the primary motivation and goal of developing the DG finite element differential calculus is to solve partial differential equations. It is shown that several existing finite element, finite difference and DG methods can be rewritten compactly using the proposed DG finite element differential calculus framework. Moreover, new DG methods for linear and nonlinear PDEs are also obtained from the framework.; Comment: 2 tables and 1 figure

An outline of possible pre-course diagnostics for differential calculus

Maharaj,Aneshkumar; Wagh,Vivek
Fonte: South African Journal of Science Publicador: South African Journal of Science
Tipo: Artigo de Revista Científica Formato: text/html
Publicado em 01/08/2014 EN
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There is a view that many first-year students lack the basic knowledge and skills expected of them to study at university level. We examined the expected work habits and pre-course diagnostics for students who choose to take a course on differential calculus. We focused on the lecturer pre-course expectations of a student in the context of work habits, knowledge and technical skills. In particular, we formulated outcomes and then sample diagnostic questions to test whether the identified learning outcomes on expected work habits and learning are in place. If students are made aware of the expected learning outcomes and if they take the diagnostic test, they should be able to achieve greater success in their studies. The validity of this assumption will be the subject of a future paper which will report on the implementation of the learning outcomes and diagnostic questions that we formulated for pre-course diagnostics in differential calculus.