Calculus III: New Applications
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en-us2017 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemThu, 25 May 2017 08:43:59 GMTThu, 25 May 2017 08:43:59 GMTNew applications in the Calculus III categoryhttp://www.mapleprimes.com/images/mapleapps.gifCalculus III: New Applications
http://www.maplesoft.com/applications/category.aspx?cid=177
Classroom Tips and Techniques: Roles for the Laplace Transform's Shifting Laws
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The shifting laws for the Laplace transform are examined, and the argument is made that the transform of f(t) Heaviside(t - a) should be done with the third shifting law, reserving the second shifting law strictly for inverting functions of the form e^(-a s) F(s). It is needlessly complicated to apply the second shifting law to functions of the form f(t) Heaviside(t - a)<img src="/view.aspx?si=1723/laplace.PNG" alt="Classroom Tips and Techniques: Roles for the Laplace Transform's Shifting Laws" align="left"/>The shifting laws for the Laplace transform are examined, and the argument is made that the transform of f(t) Heaviside(t - a) should be done with the third shifting law, reserving the second shifting law strictly for inverting functions of the form e^(-a s) F(s). It is needlessly complicated to apply the second shifting law to functions of the form f(t) Heaviside(t - a)1723Tue, 25 Apr 2017 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom 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: An Inequality-Constrained Optimization Problem
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<p>This article shows how to work both analytically and numerically to find the global maximum of</p>
<p><em>w</em> = ƒ(<em>x, y, z</em>) ≡ <em>x</em><sup>2</sup>(1 + <em>x</em>) + <em>y</em><sup>2</sup>(1 + <em>y</em>) + z<sup>2</sup>(1 + <em>z</em>)</p>
<p>in that part of the first octant on, or below, the plane <em>x</em> + <em>y</em> + <em>z</em> = 6.</p><img src="/view.aspx?si=135904/thumb.jpg" alt="Classroom Tips and Techniques: An Inequality-Constrained Optimization Problem" align="left"/><p>This article shows how to work both analytically and numerically to find the global maximum of</p>
<p><em>w</em> = ƒ(<em>x, y, z</em>) ≡ <em>x</em><sup>2</sup>(1 + <em>x</em>) + <em>y</em><sup>2</sup>(1 + <em>y</em>) + z<sup>2</sup>(1 + <em>z</em>)</p>
<p>in that part of the first octant on, or below, the plane <em>x</em> + <em>y</em> + <em>z</em> = 6.</p>135904Mon, 16 Jul 2012 04:00:00 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 LopezPhénomène de Runge - subdivision de Chebychev
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<p>On observe d'abord la divergence du polynôme de Lagrange interpolant la fonction densité de probabilité de la loi de Cauchy lorsque la <strong>subdivision est équirépartie</strong> sur [-1;1]. C'est le <u>phénomène de Runge</u>.<br />
<br />
On observe ensuite qu'en choisissant une <strong>subdivision de Chebychev</strong> le phénomène de divergence au voisinage des bornes disparait.<br />
<br />
Cette activité a été réalisé dans le cadre de la préparation à l'agrégation interne de mathématiques de Rennes le 10 Mars 2010.<br />
Les nouveaux programmes du concours incitent à proposer des exercices utilisant les TICE. Il semble difficile de proposer une preuve convaincante du phénomène de Runge pour une épreuve orale. Ceci justifie de ne s'en tenir qu'à la seule observation.</p><img src="/view.aspx?si=35301/thumb.jpg" alt="Phénomène de Runge - subdivision de Chebychev" align="left"/><p>On observe d'abord la divergence du polynôme de Lagrange interpolant la fonction densité de probabilité de la loi de Cauchy lorsque la <strong>subdivision est équirépartie</strong> sur [-1;1]. C'est le <u>phénomène de Runge</u>.<br />
<br />
On observe ensuite qu'en choisissant une <strong>subdivision de Chebychev</strong> le phénomène de divergence au voisinage des bornes disparait.<br />
<br />
Cette activité a été réalisé dans le cadre de la préparation à l'agrégation interne de mathématiques de Rennes le 10 Mars 2010.<br />
Les nouveaux programmes du concours incitent à proposer des exercices utilisant les TICE. Il semble difficile de proposer une preuve convaincante du phénomène de Runge pour une épreuve orale. Ceci justifie de ne s'en tenir qu'à la seule observation.</p>35301Fri, 26 Mar 2010 04:00:00 ZKERNIVINEN SebastienKERNIVINEN SebastienClassroom Tips and Techniques: Visualizing Regions of Integration
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<p>In this month's article, the synergy between the visual and the analytic is demonstrated with a learning tool built with Maple's embedded components.</p><img src="/view.aspx?si=34062/thumb.jpg" alt="Classroom Tips and Techniques: Visualizing Regions of Integration" align="left"/><p>In this month's article, the synergy between the visual and the analytic is demonstrated with a learning tool built with Maple's embedded components.</p>34062Wed, 21 Oct 2009 04:00:00 ZDr. Robert LopezDr. Robert LopezOptimal Portfolio Allocation and Economic Utility
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<p>Show how the risk and reward set of a portfolio of two or three securities whose returns are jointly normally distributed is calculated from standard theorems about the mean and variance of linear combinations of the random variables. Use ideas from multivariable calculus to show how the feasible set is derived. Visualize the efficient frontier of the risk and reward set. Visualize the economic utility of the portfolio. Show how the concept of economic utility selects a unique optimal portfolio on the efficient frontier.</p><img src="/view.aspx?si=4860/thumb2.jpg" alt="Optimal Portfolio Allocation and Economic Utility" align="left"/><p>Show how the risk and reward set of a portfolio of two or three securities whose returns are jointly normally distributed is calculated from standard theorems about the mean and variance of linear combinations of the random variables. Use ideas from multivariable calculus to show how the feasible set is derived. Visualize the efficient frontier of the risk and reward set. Visualize the economic utility of the portfolio. Show how the concept of economic utility selects a unique optimal portfolio on the efficient frontier.</p>4860Mon, 14 Sep 2009 04:00:00 ZShengjie GuoShengjie GuoTeorema de Cambio de Variables
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Una herramienta fundamental a la hora de trabajar con integrales múltiples es el uso del teorema de cambio de variables. A continuación, veremos de donde surge este teorema para integrales dobles y lo aplicaremos a problemas comunes de integrales múltiples.<img src="/view.aspx?si=7252/1.jpg" alt="Teorema de Cambio de Variables" align="left"/>Una herramienta fundamental a la hora de trabajar con integrales múltiples es el uso del teorema de cambio de variables. A continuación, veremos de donde surge este teorema para integrales dobles y lo aplicaremos a problemas comunes de integrales múltiples.7252Wed, 18 Feb 2009 00:00:00 ZSebastián Varas KittelSebastián Varas KittelFluid Flow Past a Cylinder
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This worksheet uses Maple to illustrate the fluid dynamics problem of flow past a cylinder.<img src="/view.aspx?si=6728/thumb.gif" alt="Fluid Flow Past a Cylinder" align="left"/>This worksheet uses Maple to illustrate the fluid dynamics problem of flow past a cylinder.6728Fri, 03 Oct 2008 00:00:00 ZMaplesoftMaplesoftStreamlines in 2-Dimensional Vector Fields
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This worksheet gives two examples of Maple's capabilities for calculating and displaying the streamlines in a 2-dimensional vector field.<img src="/view.aspx?si=6665/thumb.gif" alt="Streamlines in 2-Dimensional Vector Fields" align="left"/>This worksheet gives two examples of Maple's capabilities for calculating and displaying the streamlines in a 2-dimensional vector field.6665Tue, 16 Sep 2008 00:00:00 ZMaplesoftMaplesoftMultivariable Calculus
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This is a set of notes to accompany a course in multivariable calculus that have been developed over several years of teaching this subject.
The solution manual for these notes are available in the zip file that can be downloaded.<img src="/view.aspx?si=5241/thumb.jpg" alt="Multivariable Calculus" align="left"/>This is a set of notes to accompany a course in multivariable calculus that have been developed over several years of teaching this subject.
The solution manual for these notes are available in the zip file that can be downloaded.5241Mon, 01 Oct 2007 00:00:00 ZProf. Carl MadiganProf. Carl MadiganClassroom Tips and Techniques: Eigenvalue Problems for ODEs - Part 2
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In this second of a three-part article, we use Maple to separate variables in a partial differential equation, then show how to obtain a solution of the resulting Sturm-Liouville eigenvalue problem. The PDE is Laplace's equation in a cylinder, and separation of variables leads to an eigenvalue problem involving Bessel's equation. We solve this singular Sturm-Liouville eigenvalue problem for the two cases of homogeneous Dirichlet and Neumann boundary conditions, and for the two cases of axial symmetry and asymmetry.<img src="/view.aspx?si=5055/EigenvalueProblemsforODEs-Part2_105.jpg" alt="Classroom Tips and Techniques: Eigenvalue Problems for ODEs - Part 2" align="left"/>In this second of a three-part article, we use Maple to separate variables in a partial differential equation, then show how to obtain a solution of the resulting Sturm-Liouville eigenvalue problem. The PDE is Laplace's equation in a cylinder, and separation of variables leads to an eigenvalue problem involving Bessel's equation. We solve this singular Sturm-Liouville eigenvalue problem for the two cases of homogeneous Dirichlet and Neumann boundary conditions, and for the two cases of axial symmetry and asymmetry.5055Mon, 25 Jun 2007 00:00:00 ZDr. Robert LopezDr. Robert LopezLearning Calculus III from Examples
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The files include main topics in Calculus II: Space Geometry, Vector functions, Partial derivatives, Multiple Integrals, Vector Calculus. Many example show how to use Maple to in these topics.<img src="/view.aspx?si=4965/MultipleIntegrals_50.jpg" alt="Learning Calculus III from Examples" align="left"/>The files include main topics in Calculus II: Space Geometry, Vector functions, Partial derivatives, Multiple Integrals, Vector Calculus. Many example show how to use Maple to in these topics.4965Wed, 23 May 2007 00:00:00 ZDr. Jianzhong WangDr. Jianzhong WangClassroom Tips and Techniques:Teaching Fourier Series with Maple - Part 3
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<p>The FourierSeries package by Professor Wilhalm Werner is described. It was the first such package available on the Maple Application Center, and it has recently been updated to run in the latest versions of Maple. This article continues earlier discussions that showed how to implement, in Maple itself, the calculations related to Fourier series, then described the Fourier Series package submitted to the Maple Application Center by Professor Amir Khanshan.</p><img src="/view.aspx?si=4885/TeachingFourierSerieswithMaple-Part3_img.jpg" alt="Classroom Tips and Techniques:Teaching Fourier Series with Maple - Part 3" align="left"/><p>The FourierSeries package by Professor Wilhalm Werner is described. It was the first such package available on the Maple Application Center, and it has recently been updated to run in the latest versions of Maple. This article continues earlier discussions that showed how to implement, in Maple itself, the calculations related to Fourier series, then described the Fourier Series package submitted to the Maple Application Center by Professor Amir Khanshan.</p>4885Mon, 26 Mar 2007 04:00:00 ZDr. Robert LopezDr. Robert LopezCalculus III: Complete Set of Lessons
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A collection of 37 worksheets for 3rd semester (multivariable calculus). Developed by Fr. Mike May and Dr. Russell Blyth, St. Louis University. Topics include introductory worksheets, multivariable limits, bivariate Taylor series, Lagrange multipliers, vector and gradient fields, visualizing regions of integration, line and flux integrals.
At Saint Louis University, our Calculus III course is being taught in a computer classroom where the students have access to Maple.
The strategy I have used for bringing Maple into the classroom is to introduce it through carefully designed worksheets, which I use as:
- Lecture aids with the instructor running the worksheet with a projection system.
- Handouts for the students
- Lab assignment that the class will start together as a substitute for a lecture.
- Supplemental homework assignments.
These worksheets include a significant amount of exploratory text and exercises. The exercises ask the student to repeat the examples in the worksheets with minor modification. I do not expect them to produce the code, but rather to copy and modify a code template, focusing on the results of the problems.<img src="/view.aspx?si=4740/calcIII.gif" alt="Calculus III: Complete Set of Lessons" align="left"/>A collection of 37 worksheets for 3rd semester (multivariable calculus). Developed by Fr. Mike May and Dr. Russell Blyth, St. Louis University. Topics include introductory worksheets, multivariable limits, bivariate Taylor series, Lagrange multipliers, vector and gradient fields, visualizing regions of integration, line and flux integrals.
At Saint Louis University, our Calculus III course is being taught in a computer classroom where the students have access to Maple.
The strategy I have used for bringing Maple into the classroom is to introduce it through carefully designed worksheets, which I use as:
- Lecture aids with the instructor running the worksheet with a projection system.
- Handouts for the students
- Lab assignment that the class will start together as a substitute for a lecture.
- Supplemental homework assignments.
These worksheets include a significant amount of exploratory text and exercises. The exercises ask the student to repeat the examples in the worksheets with minor modification. I do not expect them to produce the code, but rather to copy and modify a code template, focusing on the results of the problems.4740Fri, 29 Dec 2006 00:00:00 ZProf. Michael MayProf. Michael MayClassroom Tips and Techniques: Task Templates in Maple
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Maple comes with more than 200 built-in Task Templates. The process of creating a new Task Template, and adding it to the Table of Contents of all such tasks is relatively straightforward. In this article, we explain how to create a new Task Template and add it to the list of built-in tasks.<img src="/view.aspx?si=1763/tasktemplates.gif" alt="Classroom Tips and Techniques: Task Templates in Maple" align="left"/>Maple comes with more than 200 built-in Task Templates. The process of creating a new Task Template, and adding it to the Table of Contents of all such tasks is relatively straightforward. In this article, we explain how to create a new Task Template and add it to the list of built-in tasks.1763Thu, 20 Jul 2006 00:00:00 ZDr. Robert LopezDr. Robert LopezThe Paraboloid, a Case Study
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The Paraboloid is a geometric figure very used in the engineering design. For this reason, it is important to know certain geometric characteristics like their area and volume besides the flow of a vectorial field that can cross this surface. In this worsheet, we will calculate these characteristics through double and triple integrals in different coordinate systems using the Maple commands.<img src="/view.aspx?si=1737/paraboloid.jpg" alt="The Paraboloid, a Case Study" align="left"/>The Paraboloid is a geometric figure very used in the engineering design. For this reason, it is important to know certain geometric characteristics like their area and volume besides the flow of a vectorial field that can cross this surface. In this worsheet, we will calculate these characteristics through double and triple integrals in different coordinate systems using the Maple commands.1737Mon, 29 May 2006 00:00:00 ZProf. David Macias FerrerProf. David Macias FerrerVolume of Geometric Solids through Double Integrals
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The goals of this workseet is to calculate the volume of certain geometric solids in 3D space through double integrals and to show the powerful Maple 8 graphics tools to visualize the geometric solids involved in the evaluation of double integrals.<img src="/view.aspx?si=1726/geosolid.jpg" alt="Volume of Geometric Solids through Double Integrals" align="left"/>The goals of this workseet is to calculate the volume of certain geometric solids in 3D space through double integrals and to show the powerful Maple 8 graphics tools to visualize the geometric solids involved in the evaluation of double integrals.1726Thu, 20 Apr 2006 00:00:00 ZProf. David Macias FerrerProf. David Macias FerrerAnimation: the Directional Derivative & Gradient
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Given a function of two variables f(x,y) and a domain point (x0,y0), this worksheet provides a plotting tool that can be used to plot the function f(x,y) superimposed on a plot of the line that is tangent to the graphed surface at the point (x0,y0,f(x0,y0)) and lies in a vertically oriented cutting plane determined by the direction in which the directional derivative is desired. The gradient vector is plotted in a plane parallel to the xy-coordinate plane along with a vector representing the direction associated with the directional derivative. The user has a choice of generating a single rotatable 3D plot or an animation consisting of multiple plots that show the effects of progressively changing the directions associated with the directional derivative.<img src="/view.aspx?si=1700/DirDerGra_edited.JPG" alt="Animation: the Directional Derivative & Gradient" align="left"/>Given a function of two variables f(x,y) and a domain point (x0,y0), this worksheet provides a plotting tool that can be used to plot the function f(x,y) superimposed on a plot of the line that is tangent to the graphed surface at the point (x0,y0,f(x0,y0)) and lies in a vertically oriented cutting plane determined by the direction in which the directional derivative is desired. The gradient vector is plotted in a plane parallel to the xy-coordinate plane along with a vector representing the direction associated with the directional derivative. The user has a choice of generating a single rotatable 3D plot or an animation consisting of multiple plots that show the effects of progressively changing the directions associated with the directional derivative.1700Thu, 05 Jan 2006 00:00:00 ZWilliam RichardsonWilliam Richardson