Vector Calculus: New Applications
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en-us2015 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemSat, 23 May 2015 11:59:58 GMTSat, 23 May 2015 11:59:58 GMTNew applications in the Vector Calculus categoryhttp://www.mapleprimes.com/images/mapleapps.gifVector Calculus: New Applications
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Descartes & Mme La Marquise du Chatelet And The Elastic Collision of Two Bodies
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<p><strong><em> ABSTRACT<br /> <br /> The Marquise</em></strong> <strong><em>du Chatelet in her book " Les Institutions Physiques" published in 1740, stated on page 36, that Descartes, when formulating his laws of motion in an elastic collision of two bodies B & C (B being more massive than C) <span >having the same speed v</span>, said that t<span >he smaller one C will reverse its course </span>while <span >the more massive body B will continue its course in the same direction as before</span> and <span >both will have again the same speed v.<br /> <br /> </span>Mme du Chatelet, basing her judgment on theoretical considerations using <span >the principle of continuity</span> , declared that Descartes was <span >wrong</span> in his statement. For Mme du Chatelet the larger mass B should reverse its course and move in the opposite direction. She mentioned nothing about both bodies B & C as <span >having the same velocity after collision as Descartes did</span>.<br /> <br /> At the time of Descartes, some 300 years ago, the concept of kinetic energy & momentum as we know today was not yet well defined, let alone considered in any physical problem.<br /> <br /> Actually both Descartes & Mme du Chatelet may have been right in some special cases but not in general as the discussion that follows will show.</em></strong></p><img src="/view.aspx?si=153515/Elastic_Collision_image1.jpg" alt="Descartes & Mme La Marquise du Chatelet And The Elastic Collision of Two Bodies" align="left"/><p><strong><em> ABSTRACT<br /> <br /> The Marquise</em></strong> <strong><em>du Chatelet in her book " Les Institutions Physiques" published in 1740, stated on page 36, that Descartes, when formulating his laws of motion in an elastic collision of two bodies B & C (B being more massive than C) <span >having the same speed v</span>, said that t<span >he smaller one C will reverse its course </span>while <span >the more massive body B will continue its course in the same direction as before</span> and <span >both will have again the same speed v.<br /> <br /> </span>Mme du Chatelet, basing her judgment on theoretical considerations using <span >the principle of continuity</span> , declared that Descartes was <span >wrong</span> in his statement. For Mme du Chatelet the larger mass B should reverse its course and move in the opposite direction. She mentioned nothing about both bodies B & C as <span >having the same velocity after collision as Descartes did</span>.<br /> <br /> At the time of Descartes, some 300 years ago, the concept of kinetic energy & momentum as we know today was not yet well defined, let alone considered in any physical problem.<br /> <br /> Actually both Descartes & Mme du Chatelet may have been right in some special cases but not in general as the discussion that follows will show.</em></strong></p>153515Fri, 07 Mar 2014 05:00:00 ZDr. Ahmed BaroudyDr. Ahmed BaroudyCollision 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/Collision_Detection_image1.jpg" 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 LopezClassroom Tips and Techniques: New Tools for Lines and Planes
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The fifteen new "Lines and Planes" commands in the Student MultivariateCalculus package are detailed, and then illustrated via a collection of examples from a typical calculus course. These new commands can also be implemented through the Context Menu system, as shown by parallel solutions in the set of examples.<img src="/view.aspx?si=144642/thumb.jpg" alt="Classroom Tips and Techniques: New Tools for Lines and Planes" align="left"/>The fifteen new "Lines and Planes" commands in the Student MultivariateCalculus package are detailed, and then illustrated via a collection of examples from a typical calculus course. These new commands can also be implemented through the Context Menu system, as shown by parallel solutions in the set of examples.144642Thu, 14 Mar 2013 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Animated Trace of a Curve Drawn by Radius Vector
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A plane curve <strong>R</strong>(<em>t</em>) = <em>x</em>(<em>t</em>) <strong>i</strong> + <em>y</em>(<em>t</em>) <strong>j</strong> is traced by a "moving" radius vector <strong>R</strong>(<em>t</em>). Code for this animation is explored in this article.<img src="/view.aspx?si=143371/thumb.jpg" alt="Classroom Tips and Techniques: Animated Trace of a Curve Drawn by Radius Vector" align="left"/>A plane curve <strong>R</strong>(<em>t</em>) = <em>x</em>(<em>t</em>) <strong>i</strong> + <em>y</em>(<em>t</em>) <strong>j</strong> is traced by a "moving" radius vector <strong>R</strong>(<em>t</em>). Code for this animation is explored in this article.143371Mon, 11 Feb 2013 05:00:00 ZDr. Robert LopezDr. Robert LopezClassroom 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. Practicum. Analytic Geometry.
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<p>Examples. 2 x 30 variants.<br />The main thing - the idea.<br />To activate the solution, press the appropriate button, and then - Enter or !!!</p><img src="/view.aspx?si=102180/ag1.jpg" alt="Mapler. Practicum. Analytic Geometry." align="left"/><p>Examples. 2 x 30 variants.<br />The main thing - the idea.<br />To activate the solution, press the appropriate button, and then - Enter or !!!</p>102180Thu, 03 Mar 2011 05:00:00 ZDonetsk National UniversityDonetsk National UniversityClassroom Tips and Techniques: More Gems from the Little Red Book of Maple Magic
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Five more bits of "Maple magic" accumulated in recent months are shared: "if" with certain exact numbers, constant functions, replacing a product with a name, assumptions on subscripted variables, and gradient vectors via the Matrix palette.<img src="/view.aspx?si=101922/thumb.jpg" alt="Classroom Tips and Techniques: More Gems from the Little Red Book of Maple Magic" align="left"/>Five more bits of "Maple magic" accumulated in recent months are shared: "if" with certain exact numbers, constant functions, replacing a product with a name, assumptions on subscripted variables, and gradient vectors via the Matrix palette.101922Tue, 22 Feb 2011 05:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Gems from the Little Red Book of Maple Magic
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Five bits of "Maple magic" accumulated in recent months are shared: converting the half-angle trig formulas to radicals, tickmarks along a parametric curve, writing unevaluated math on a graph, changing Maple's differentiation formulas, and drawing a decent surface for a function containing a square root.<img src="/view.aspx?si=100897/thumb.jpg" alt="Classroom Tips and Techniques: Gems from the Little Red Book of Maple Magic" align="left"/>Five bits of "Maple magic" accumulated in recent months are shared: converting the half-angle trig formulas to radicals, tickmarks along a parametric curve, writing unevaluated math on a graph, changing Maple's differentiation formulas, and drawing a decent surface for a function containing a square root.100897Fri, 14 Jan 2011 05:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Partial Derivatives by Subscripting
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As output, Maple can display the partial derivative ∂/∂<em>x f</em>(<em>x,y</em>) as <em>f</em><sub>x</sub>; that is, subscript notation can be used to display partial derivatives, and it can be done with two completely different mechanisms. This article describes these two techniques, and then investigates the extent to which partial derivatives can be calculated by subscript notation.<img src="/view.aspx?si=100266/thumb.jpg" alt="Classroom Tips and Techniques: Partial Derivatives by Subscripting" align="left"/>As output, Maple can display the partial derivative ∂/∂<em>x f</em>(<em>x,y</em>) as <em>f</em><sub>x</sub>; that is, subscript notation can be used to display partial derivatives, and it can be done with two completely different mechanisms. This article describes these two techniques, and then investigates the extent to which partial derivatives can be calculated by subscript notation.100266Wed, 15 Dec 2010 05:00:00 ZDr. Robert LopezDr. Robert LopezClassroom 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: Geodesics on a Surface
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<p>Several months ago we provided the article Tensor Calculus with the Differential Geometry Package in which we found geodesics in the plane when the plane was referred to polar coordinates. In this month's article we find geodesics on a surface embedded in R<sup>3</sup>. We illustrate three approaches: numeric approximation, the calculus of variations, and differential geometry.</p><img src="/view.aspx?si=34940/thumb.jpg" alt="Classroom Tips and Techniques: Geodesics on a Surface" align="left"/><p>Several months ago we provided the article Tensor Calculus with the Differential Geometry Package in which we found geodesics in the plane when the plane was referred to polar coordinates. In this month's article we find geodesics on a surface embedded in R<sup>3</sup>. We illustrate three approaches: numeric approximation, the calculus of variations, and differential geometry.</p>34940Tue, 08 Dec 2009 05:00:00 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 Lopez