Vector Calculus: New Applications
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en-us2017 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemTue, 17 Jan 2017 10:56:42 GMTTue, 17 Jan 2017 10:56:42 GMTNew applications in the Vector Calculus categoryhttp://www.mapleprimes.com/images/mapleapps.gifVector Calculus: New Applications
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Vectors in the plane.
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If an object is subjected to several forces having different magnitudes and act in different directions, how can determine the magnitude and direction of the resultant total force on the object? Forces are vectors and should be added according to the definition of the vector sum. Engineering dealing with many quantities that have both magnitude and direction and can be expressed and analyzed as vectors.
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In Spanish.<img src="/view.aspx?si=154071/vpThumb.jpg" alt="Vectors in the plane." align="left"/>If an object is subjected to several forces having different magnitudes and act in different directions, how can determine the magnitude and direction of the resultant total force on the object? Forces are vectors and should be added according to the definition of the vector sum. Engineering dealing with many quantities that have both magnitude and direction and can be expressed and analyzed as vectors.
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In Spanish.154071Fri, 01 Apr 2016 04:00:00 ZProf. Lenin Araujo CastilloProf. Lenin Araujo CastilloCollision detection between toolholder and workpiece on ball nut grinding
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<p>In this worksheet a collision detection performed to determine the minimum safety distance between a tool holder and ball nut on grinding manufacturing. A nonlinear quartic equation system have to be solved by <em>Newton's</em> and <em>Broyden's</em> methods and results are compared with <em>Maple fsolve()</em> command. Users can check the different results by embedded components and animated 3D surface plot.</p><img src="/view.aspx?si=153477/0320a66eb812382755a045a5251b1390.gif" alt="Collision detection between toolholder and workpiece on ball nut grinding" align="left"/><p>In this worksheet a collision detection performed to determine the minimum safety distance between a tool holder and ball nut on grinding manufacturing. A nonlinear quartic equation system have to be solved by <em>Newton's</em> and <em>Broyden's</em> methods and results are compared with <em>Maple fsolve()</em> command. Users can check the different results by embedded components and animated 3D surface plot.</p>153477Mon, 23 Dec 2013 05:00:00 ZGyörgy HegedûsGyörgy HegedûsClassroom Tips and Techniques: Drawing a Normal and Tangent Plane on a Surface
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Four different techniques are given for obtaining a graph showing a surface with a normal and tangent plane attached. The work is a response to <a href="http://www.mapleprimes.com/questions/147681-A-Problem-About-Plot-The-Part-Of-The-Surface">a MaplePrimes question asked on May 25, 2013</a>.<img src="/view.aspx?si=150722/thumb.jpg" alt="Classroom Tips and Techniques: Drawing a Normal and Tangent Plane on a Surface" align="left"/>Four different techniques are given for obtaining a graph showing a surface with a normal and tangent plane attached. The work is a response to <a href="http://www.mapleprimes.com/questions/147681-A-Problem-About-Plot-The-Part-Of-The-Surface">a MaplePrimes question asked on May 25, 2013</a>.150722Tue, 20 Aug 2013 04:00:00 ZDr. Robert LopezDr. Robert LopezzoMbi
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<p>Higher Mathematics for external students of biological faculty.<br />Solver-practicum.<br />1st semester.<br />300 problems (15 labs in 20 variants).<br />mw.zip</p>
<p>Before use - Shake! <br />(Click on the button and activate the program and Maplet).<br />Full version in html: <a href="http://webmath.exponenta.ru/zom/index.html">http://webmath.exponenta.ru/zom/index.html</a></p><img src="/view.aspx?si=129642/zombie_3.jpg" alt="zoMbi" align="left"/><p>Higher Mathematics for external students of biological faculty.<br />Solver-practicum.<br />1st semester.<br />300 problems (15 labs in 20 variants).<br />mw.zip</p>
<p>Before use - Shake! <br />(Click on the button and activate the program and Maplet).<br />Full version in html: <a href="http://webmath.exponenta.ru/zom/index.html">http://webmath.exponenta.ru/zom/index.html</a></p>129642Sun, 15 Jan 2012 05:00:00 ZDr. Valery CyboulkoDr. Valery CyboulkoClassroom Tips and Techniques: Directional Derivatives in Maple
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Several identities in vector calculus involve the operator A . (VectorCalculus[Nabla]) acting on a vector B. The resulting expression (A . (VectorCalculus[Nabla]))B is interpreted as the directional derivative of the vector B in the direction of the vector A. This is not easy to implement in Maple's VectorCalculus packages. However, this functionality exists in the Physics:-Vectors package, and in the DifferentialGeometry package where it is properly called the DirectionalCovariantDerivative.
This article examines how to obtain (A . (VectorCalculus[Nabla]))B in Maple.<img src="/view.aspx?si=126623/thumb.jpg" alt="Classroom Tips and Techniques: Directional Derivatives in Maple" align="left"/>Several identities in vector calculus involve the operator A . (VectorCalculus[Nabla]) acting on a vector B. The resulting expression (A . (VectorCalculus[Nabla]))B is interpreted as the directional derivative of the vector B in the direction of the vector A. This is not easy to implement in Maple's VectorCalculus packages. However, this functionality exists in the Physics:-Vectors package, and in the DifferentialGeometry package where it is properly called the DirectionalCovariantDerivative.
This article examines how to obtain (A . (VectorCalculus[Nabla]))B in Maple.126623Fri, 14 Oct 2011 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Gems 16-20 from the Red Book of Maple Magic
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From the Red Book of Maple Magic, Gems 16-20: Vectors with assumptions in VectorCalculus, aliasing commands to symbols, setting iterated integrals from the Expression palette, writing a slider value to a label, and writing text to a math container.<img src="/view.aspx?si=125886/thumb.jpg" alt="Classroom Tips and Techniques: Gems 16-20 from the Red Book of Maple Magic" align="left"/>From the Red Book of Maple Magic, Gems 16-20: Vectors with assumptions in VectorCalculus, aliasing commands to symbols, setting iterated integrals from the Expression palette, writing a slider value to a label, and writing text to a math container.125886Fri, 23 Sep 2011 04:00:00 ZDr. Robert LopezDr. Robert LopezMapler. 05. Аlgebraic equations & Index
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<p>Mathematical program-controlled multivariate Workshop.<br />Version without maplets and test problems. <br />Further depends on community interest.</p><img src="/view.aspx?si=102285/mrs.jpg" alt="Mapler. 05. Аlgebraic equations & Index" align="left"/><p>Mathematical program-controlled multivariate Workshop.<br />Version without maplets and test problems. <br />Further depends on community interest.</p>102285Mon, 07 Mar 2011 05:00:00 ZDr. Valery CyboulkoDr. Valery CyboulkoExotic EIE-course
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<p>Ukraine. <br />Exotic training course for the entrance examination in mathematics.<br /><strong>External independent evaluation</strong> <br />Themes:<br />0101 Goals and rational number <br />0102 Interest. The main problem of interest <br />0103 The simplest geometric shapes on the plane and their properties <br />0201 Degree of natural and integral indicator <br />0202 Monomial and polynomials and operations on them <br />0203 Triangles and their basic properties <br />0301 Algebraic fractions and operations on them <br />0302 Square root. Real numbers <br />0303 Circle and circle, their properties <br />0401 Equations, inequalities and their systems <br />0402 Function and its basic properties <br />0403 Described and inscribed triangles <br />0501 Linear function, linear equations, inequalities and their systems <br />0502 Quadratic function, quadratic equation, inequality and their systems <br />0503 Solving square triangles <br />0601 Rational Equations, Inequalities and their sysytemy <br />0602 Numerical sequence. Arithmetic and geometric progression <br />0603 Solving arbitrary triangles <br />0701 Sine, cosine, tangent and cotangent numeric argument <br />0702 Identical transformation of trigonometric expressions <br />0703 Quadrilateral types and their basic properties <br />0801 Trigonometric and inverse trigonometric functions, their properties <br />0802 Trigonometric equations and inequalities <br />0803 Polygons and their properties <br />0901 The root of n-th degree. Degree of rational parameters <br />0902 The power functions and their properties. Irrational equations, inequalities and their systems <br />0903 Regular polygons and their properties <br />1001 Logarithms. Logarithmic function. Logarithmic equations, inequalities and their systems <br />1002 Exponential function. Indicator of equations, inequalities and their systems <br />1003 Direct and planes in space <br />1101 Derivative and its geometric and mechanical content <br />1102 Derivatives and its application <br />1103 Polyhedron. Prisms and pyramids. Regular polyhedron <br />1201 Initial and definite integral <br />1202 Application of certain integral <br />1203 Body rotation <br />1301 Compounds. Binomial theorem <br />1302 General methods for solving equations, inequalities and their systems <br />1303 Coordinates in the plane and in space <br />1401 The origins of probability theory <br />1402 Beginnings of Mathematical Statistics <br />1403 Vectors in the plane and in space <br /><strong>Maple </strong>version<br /><strong>Html-interactive</strong> version</p><img src="/view.aspx?si=102076/ell.jpg" alt="Exotic EIE-course" align="left"/><p>Ukraine. <br />Exotic training course for the entrance examination in mathematics.<br /><strong>External independent evaluation</strong> <br />Themes:<br />0101 Goals and rational number <br />0102 Interest. The main problem of interest <br />0103 The simplest geometric shapes on the plane and their properties <br />0201 Degree of natural and integral indicator <br />0202 Monomial and polynomials and operations on them <br />0203 Triangles and their basic properties <br />0301 Algebraic fractions and operations on them <br />0302 Square root. Real numbers <br />0303 Circle and circle, their properties <br />0401 Equations, inequalities and their systems <br />0402 Function and its basic properties <br />0403 Described and inscribed triangles <br />0501 Linear function, linear equations, inequalities and their systems <br />0502 Quadratic function, quadratic equation, inequality and their systems <br />0503 Solving square triangles <br />0601 Rational Equations, Inequalities and their sysytemy <br />0602 Numerical sequence. Arithmetic and geometric progression <br />0603 Solving arbitrary triangles <br />0701 Sine, cosine, tangent and cotangent numeric argument <br />0702 Identical transformation of trigonometric expressions <br />0703 Quadrilateral types and their basic properties <br />0801 Trigonometric and inverse trigonometric functions, their properties <br />0802 Trigonometric equations and inequalities <br />0803 Polygons and their properties <br />0901 The root of n-th degree. Degree of rational parameters <br />0902 The power functions and their properties. Irrational equations, inequalities and their systems <br />0903 Regular polygons and their properties <br />1001 Logarithms. Logarithmic function. Logarithmic equations, inequalities and their systems <br />1002 Exponential function. Indicator of equations, inequalities and their systems <br />1003 Direct and planes in space <br />1101 Derivative and its geometric and mechanical content <br />1102 Derivatives and its application <br />1103 Polyhedron. Prisms and pyramids. Regular polyhedron <br />1201 Initial and definite integral <br />1202 Application of certain integral <br />1203 Body rotation <br />1301 Compounds. Binomial theorem <br />1302 General methods for solving equations, inequalities and their systems <br />1303 Coordinates in the plane and in space <br />1401 The origins of probability theory <br />1402 Beginnings of Mathematical Statistics <br />1403 Vectors in the plane and in space <br /><strong>Maple </strong>version<br /><strong>Html-interactive</strong> version</p>102076Mon, 28 Feb 2011 05:00:00 ZTIMOTIMOClassroom Tips and Techniques: Stepwise Solutions in Maple - Part 3 - Vector Calculus
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<p>In our previous two articles we have discussed the commands, Assistants, Tutors, and Task Templates that implement stepwise calculations in algebra, calculus (both of one and several variables) and linear algebra. In this month's article, we illustrate the stepwise tools available in vector calculus.</p><img src="/view.aspx?si=35359/thumb.jpg" alt="Classroom Tips and Techniques: Stepwise Solutions in Maple - Part 3 - Vector Calculus" align="left"/><p>In our previous two articles we have discussed the commands, Assistants, Tutors, and Task Templates that implement stepwise calculations in algebra, calculus (both of one and several variables) and linear algebra. In this month's article, we illustrate the stepwise tools available in vector calculus.</p>35359Fri, 09 Apr 2010 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Point-and-Click Access to the Differential Operators of Vector Calculus
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<p>This month's article describes new tools that allow a syntax-free setting of coordinate systems so that the differential operators of vector calculus can be implemented via VectorCalculus[Nabla], the "del" or "nabla" symbol.</p><img src="/view.aspx?si=33081/thumb.png" alt="Classroom Tips and Techniques: Point-and-Click Access to the Differential Operators of Vector Calculus" align="left"/><p>This month's article describes new tools that allow a syntax-free setting of coordinate systems so that the differential operators of vector calculus can be implemented via VectorCalculus[Nabla], the "del" or "nabla" symbol.</p>33081Thu, 04 Jun 2009 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Plotting a Slice of a Vector Field
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This month's article answers a user's question about plotting a "slice" of a vector field, and superimposing this graph on a density plot of the underlying scalar field. The "slice" of the vector field is a graph of the field arrows emanating from a coordinate plane.<img src="/view.aspx?si=19202/thumb.png" alt="Classroom Tips and Techniques: Plotting a Slice of a Vector Field" align="left"/>This month's article answers a user's question about plotting a "slice" of a vector field, and superimposing this graph on a density plot of the underlying scalar field. The "slice" of the vector field is a graph of the field arrows emanating from a coordinate plane.19202Fri, 06 Mar 2009 00:00:00 ZDr. Robert LopezDr. Robert LopezVisualizing the Laplace-Runge-Lenz Vector
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The vector treatment of Kepler's first law presented in most calculus textbooks is based on the existence of a constant vector that is associated with the exact inverse square force law. Such a treatment is not a general substitute for the methods based on the differential equations of motion. This worksheet demonstrates how to use Maple to solve the differential equations governing planetary motion, and how to visualize the Laplace-Runge-Lenz vector which is peculiar to the force law of the form k/r^2.<img src="/view.aspx?si=19187/thumb.gif" alt="Visualizing the Laplace-Runge-Lenz Vector" align="left"/>The vector treatment of Kepler's first law presented in most calculus textbooks is based on the existence of a constant vector that is associated with the exact inverse square force law. Such a treatment is not a general substitute for the methods based on the differential equations of motion. This worksheet demonstrates how to use Maple to solve the differential equations governing planetary motion, and how to visualize the Laplace-Runge-Lenz vector which is peculiar to the force law of the form k/r^2.19187Mon, 02 Mar 2009 00:00:00 ZDr. Frank WangDr. Frank WangClassroom Tips and Techniques: Electric Field from Distributed Charge
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The electric field from a constant line charge of finite length is computed in both the VectorCalculus and Physics packages. A field plot along with field lines is also obtained.<img src="/view.aspx?si=7217/1.jpg" alt="Classroom Tips and Techniques: Electric Field from Distributed Charge" align="left"/>The electric field from a constant line charge of finite length is computed in both the VectorCalculus and Physics packages. A field plot along with field lines is also obtained.7217Mon, 09 Feb 2009 00:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Plotting Curves Defined Parametrically
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In this article, we compare the Maple 12 options for graphing curves given parametrically in two or three dimensions.<img src="/view.aspx?si=6725/Plotting_Curves_Defined_Par.jpg" alt="Classroom Tips and Techniques: Plotting Curves Defined Parametrically" align="left"/>In this article, we compare the Maple 12 options for graphing curves given parametrically in two or three dimensions.6725Thu, 02 Oct 2008 00:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Visualizing the Plane Determined by Two Vectors at a Point in Space
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In this month's article we examine how tools from the VectorCalculus, Student[LinearAlgebra], and Physics packages help obtain and visualize the plane determined by a point and two given directions.<img src="/view.aspx?si=5571/thumb.jpg" alt="Classroom Tips and Techniques: Visualizing the Plane Determined by Two Vectors at a Point in Space" align="left"/>In this month's article we examine how tools from the VectorCalculus, Student[LinearAlgebra], and Physics packages help obtain and visualize the plane determined by a point and two given directions.5571Mon, 07 Jan 2008 00:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Shading between Curves, and Integrating Vectors Componentwise
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This article shows how to use built-in Maple functionalities to shade between two curves, and discusses modifications to the int command provided in the VectorCalculus package. In particular, the top-level paradigm where int is immediate and Int is delayed is sacrificed to the added functionalities of immediate mapping over vectors, iterated integration, and integration over predefined regions.<img src="/view.aspx?si=5466/R-27_Shading_between_Curves_and_Integrating_Vectors_Com.jpg" alt="Classroom Tips and Techniques: Shading between Curves, and Integrating Vectors Componentwise" align="left"/>This article shows how to use built-in Maple functionalities to shade between two curves, and discusses modifications to the int command provided in the VectorCalculus package. In particular, the top-level paradigm where int is immediate and Int is delayed is sacrificed to the added functionalities of immediate mapping over vectors, iterated integration, and integration over predefined regions.5466Mon, 29 Oct 2007 00:00:00 ZDr. Robert LopezDr. Robert LopezDynamics in Spherical Coordinates
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A brief introduction to dynamics in spherical coordinates.<img src="/view.aspx?si=4892/Dynamics in Spherical Coords Sketch 1.jpg" alt="Dynamics in Spherical Coordinates" align="left"/>A brief introduction to dynamics in spherical coordinates.4892Thu, 05 Apr 2007 00:00:00 ZJ. M. RedwoodJ. M. RedwoodClassroom Tips and Techniques: The Lagrange Multiplier Method
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Maple has a number of graphical and analytical tools for studying and implementing the method of Lagrange multipliers. In this article, we demonstrate a number of these tools, indicating how they might be used pedagogically.<img src="/view.aspx?si=4811/lagrange.gif" alt="Classroom Tips and Techniques: The Lagrange Multiplier Method" align="left"/>Maple has a number of graphical and analytical tools for studying and implementing the method of Lagrange multipliers. In this article, we demonstrate a number of these tools, indicating how they might be used pedagogically.4811Mon, 28 Aug 2006 00:00:00 ZDr. Robert LopezDr. Robert LopezStokes' Theorem
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There are some examples for Stokes' integral Theorem in the worksheet. One can check the Theorem by examples, in arbitrary dimensional vector space, for abitrary dimensional submanifolds, for differentable functions.<img src="/view.aspx?si=1755/stokesend_175.gif" alt="Stokes' Theorem" align="left"/>There are some examples for Stokes' integral Theorem in the worksheet. One can check the Theorem by examples, in arbitrary dimensional vector space, for abitrary dimensional submanifolds, for differentable functions.1755Mon, 26 Jun 2006 00:00:00 ZAttila AndaiAttila AndaiProcedimientos Para Integrar Campos Vectoriales
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Este trabajo presenta una forma de abordar el estudio de integrales de campos vectoriales de dos o más variables mediante el uso de MAPLE. En un curso de Cálculo de funciones de más de una variable, en carreras de Ingeniería o Ciencias de la Tierra, es una temática de gran importancia debido a sus múltiples aplicaciones, pero al mismo tiempo es de mucha dificultad para el alumno por el grado de abstracción que estos conceptos requieren.
El objetivo de esta presentación es afianzar el uso de campos vectoriales, divergencia, rotor, flujo de un vector o de un rotor, por medio de este Sistema de Cálculo Simbólico.<img src="/view.aspx?si=4500//applications/images/app_image_blank_lg.jpg" alt="Procedimientos Para Integrar Campos Vectoriales" align="left"/>Este trabajo presenta una forma de abordar el estudio de integrales de campos vectoriales de dos o más variables mediante el uso de MAPLE. En un curso de Cálculo de funciones de más de una variable, en carreras de Ingeniería o Ciencias de la Tierra, es una temática de gran importancia debido a sus múltiples aplicaciones, pero al mismo tiempo es de mucha dificultad para el alumno por el grado de abstracción que estos conceptos requieren.
El objetivo de esta presentación es afianzar el uso de campos vectoriales, divergencia, rotor, flujo de un vector o de un rotor, por medio de este Sistema de Cálculo Simbólico.4500Mon, 17 May 2004 11:40:41 ZZulma MillanZulma Millan