Dr. Robert Lopez: New Applications
http://www.maplesoft.com/applications/author.aspx?mid=1629
en-us2017 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemTue, 26 Sep 2017 20:06:25 GMTTue, 26 Sep 2017 20:06:25 GMTNew applications published by Dr. Robert Lopezhttp://www.mapleprimes.com/images/mapleapps.gifDr. Robert Lopez: New Applications
http://www.maplesoft.com/applications/author.aspx?mid=1629
Classroom Tips and Techniques: The Partial Fraction Decomposition
https://www.maplesoft.com/applications/view.aspx?SID=1753&ref=Feed
The algebraic technique of partial fraction decomposition typically appears first in the integral calculus course as part of the methodology of integrating rational functions; and second, in any course such as differential equations where Laplace transforms must be inverted. If the Laplace transform is used in an engineering course, partial fraction decompositions must generally be implemented over the complex field so that all factors are linear.
This column describes how to obtain the partial fraction decomposition in Maple, either with irreducible quadratic factors or with strictly linear factors. We also suggest some pedagogic devices for providing insight into the algebraic processes involved.<img src="/view.aspx?si=1753/partialfrac.PNG" alt="Classroom Tips and Techniques: The Partial Fraction Decomposition" align="left"/>The algebraic technique of partial fraction decomposition typically appears first in the integral calculus course as part of the methodology of integrating rational functions; and second, in any course such as differential equations where Laplace transforms must be inverted. If the Laplace transform is used in an engineering course, partial fraction decompositions must generally be implemented over the complex field so that all factors are linear.
This column describes how to obtain the partial fraction decomposition in Maple, either with irreducible quadratic factors or with strictly linear factors. We also suggest some pedagogic devices for providing insight into the algebraic processes involved.1753Fri, 25 Aug 2017 04:00:00 ZDr. Robert LopezDr. Robert LopezEigenpairs: What are they and how they are found
https://www.maplesoft.com/applications/view.aspx?SID=154291&ref=Feed
Clearly, Maple can compute eigenpairs (eigenvalues and eigenvectors) for a matrix, but of what help is Maple in getting across the concept of an eigenpair, and relating that insight to the standard algorithms students are expected to use to find them? This application is the companion Maple document to the webinar “Eigenpairs in Maple”, presented by Dr. Robert Lopez. In both the webinar and this application, he demonstrates how Maple can enhance the task of teaching the eigenpair concept, and shows how Maple bridges the gap between the concept and the algorithms.
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<B>View the Recorded Webinar:</B><BR>
<A HREF="/webinars/recorded/featured.aspx?id=1181">Eigenpairs in Maple</A><img src="/view.aspx?si=154291/eigenpair.jpg" alt="Eigenpairs: What are they and how they are found" align="left"/>Clearly, Maple can compute eigenpairs (eigenvalues and eigenvectors) for a matrix, but of what help is Maple in getting across the concept of an eigenpair, and relating that insight to the standard algorithms students are expected to use to find them? This application is the companion Maple document to the webinar “Eigenpairs in Maple”, presented by Dr. Robert Lopez. In both the webinar and this application, he demonstrates how Maple can enhance the task of teaching the eigenpair concept, and shows how Maple bridges the gap between the concept and the algorithms.
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<B>View the Recorded Webinar:</B><BR>
<A HREF="/webinars/recorded/featured.aspx?id=1181">Eigenpairs in Maple</A>154291Fri, 25 Aug 2017 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Eigenvalue Problems for ODEs
https://www.maplesoft.com/applications/view.aspx?SID=4971&ref=Feed
Some boundary value problems for partial differential equations are amenable to analytic techniques. For example, the constant-coefficient, second-order linear equations called the heat, wave, and potential equations are solved with some type of Fourier series representation obtained from the Sturm-Liouville eigenvalue problem that arises upon separating variables. The role of Maple in the solution of such boundary value problems is examined. Efficient techniques for separating variables, and a way to guide Maple through the solution of the resulting Sturm-Liouville eigenvalue problems are shown.<img src="/view.aspx?si=4971/R-23EigenvalueProblemsforODEs.jpg" alt="Classroom Tips and Techniques: Eigenvalue Problems for ODEs" align="left"/>Some boundary value problems for partial differential equations are amenable to analytic techniques. For example, the constant-coefficient, second-order linear equations called the heat, wave, and potential equations are solved with some type of Fourier series representation obtained from the Sturm-Liouville eigenvalue problem that arises upon separating variables. The role of Maple in the solution of such boundary value problems is examined. Efficient techniques for separating variables, and a way to guide Maple through the solution of the resulting Sturm-Liouville eigenvalue problems are shown.4971Mon, 14 Aug 2017 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Green's Functions for Second-Order ODEs
https://www.maplesoft.com/applications/view.aspx?SID=4820&ref=Feed
<p>For second-order ODEs, we compute the Green's function for both initial and boundary value problems. For the boundary value problem, we consider mixed and unmixed boundary conditions, of both homogeneous and nonhomogeneous types. In every case, we compare our solutions to direct solutions using Maple's dsolve command.</p><img src="/view.aspx?si=4820/image.php.gif" alt="Classroom Tips and Techniques: Green's Functions for Second-Order ODEs" align="left"/><p>For second-order ODEs, we compute the Green's function for both initial and boundary value problems. For the boundary value problem, we consider mixed and unmixed boundary conditions, of both homogeneous and nonhomogeneous types. In every case, we compare our solutions to direct solutions using Maple's dsolve command.</p>4820Tue, 04 Jul 2017 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: The Lagrange Multiplier Method
https://www.maplesoft.com/applications/view.aspx?SID=4811&ref=Feed
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.PNG" 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.4811Tue, 23 May 2017 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Roles for the Laplace Transform's Shifting Laws
https://www.maplesoft.com/applications/view.aspx?SID=1723&ref=Feed
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: Integration by Parts
https://www.maplesoft.com/applications/view.aspx?SID=1742&ref=Feed
Maple implements integration by parts with two different commands. One was designed in a pedagogical setting, and the other, for a "production" setting. In this article, we compare the functionalities of these two commands.<img src="/view.aspx?si=1742/tutor.png" alt="Classroom Tips and Techniques: Integration by Parts" align="left"/>Maple implements integration by parts with two different commands. One was designed in a pedagogical setting, and the other, for a "production" setting. In this article, we compare the functionalities of these two commands.1742Thu, 02 Mar 2017 05:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Norm of a Matrix
https://www.maplesoft.com/applications/view.aspx?SID=1430&ref=Feed
The greatest benefits from bringing Maple into the classroom are realized when the static pedagogy of a printed textbook is enlivened by the interplay of symbolic, graphic, and numeric calculations made possible by technology. Getting Maple to compute the correct answer is just the first step. Using Maple to bring insights not easily realized with by-hand calculations should be the goal of everyone who sets a hand to improving the learning experiences of students. In this article we will show how Maple can be used to gain insight on what the norm of a matrix means.<img src="/view.aspx?si=1430/thumb.jpg" alt="Classroom Tips and Techniques: Norm of a Matrix" align="left"/>The greatest benefits from bringing Maple into the classroom are realized when the static pedagogy of a printed textbook is enlivened by the interplay of symbolic, graphic, and numeric calculations made possible by technology. Getting Maple to compute the correct answer is just the first step. Using Maple to bring insights not easily realized with by-hand calculations should be the goal of everyone who sets a hand to improving the learning experiences of students. In this article we will show how Maple can be used to gain insight on what the norm of a matrix means.1430Mon, 13 Feb 2017 05:00:00 ZDr. Robert LopezDr. Robert LopezGirding the Equator of Earth with a Belt
https://www.maplesoft.com/applications/view.aspx?SID=154220&ref=Feed
This is a problem that appears in many calculus texts. The problem is that of girding the equator of the earth with a belt, then extending by one unit (here, taken as the foot) the radius of the circle so formed. The question is by how much does the circumference of the belt increase. This problem usually appears in the section of the calculus text dealing with linear approximations by the differential.
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This app is also the subject of a blog post on MaplePrimes: <A HREF="http://www.mapleprimes.com/maplesoftblog/207876-Girding-The-Equator-Of-The-Earth-With-A-Belt">Girding the Equator of Earth with a Belt</A><img src="/view.aspx?si=154220/b54e83a2917e82eae1adfaebd0567f9c.gif" alt="Girding the Equator of Earth with a Belt" align="left"/>This is a problem that appears in many calculus texts. The problem is that of girding the equator of the earth with a belt, then extending by one unit (here, taken as the foot) the radius of the circle so formed. The question is by how much does the circumference of the belt increase. This problem usually appears in the section of the calculus text dealing with linear approximations by the differential.
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This app is also the subject of a blog post on MaplePrimes: <A HREF="http://www.mapleprimes.com/maplesoftblog/207876-Girding-The-Equator-Of-The-Earth-With-A-Belt">Girding the Equator of Earth with a Belt</A>154220Tue, 07 Feb 2017 05:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Real and Complex Derivatives of Some Elementary Functions
https://www.maplesoft.com/applications/view.aspx?SID=153726&ref=Feed
The elementary functions include the six trigonometric and hyperbolic functions and their inverses. For all but five of these 24 functions, Maple's derivative (correct on the complex plane) agrees with the real-variable form found in the standard calculus text. For these five exceptions, this article explores two issues: (1) Does Maple's derivative, restricted to the real domain, agree with the real-variable form; and (2), to what extent do both forms agree on the complex plane.<img src="/view.aspx?si=153726/thumb.jpg" alt="Classroom Tips and Techniques: Real and Complex Derivatives of Some Elementary Functions" align="left"/>The elementary functions include the six trigonometric and hyperbolic functions and their inverses. For all but five of these 24 functions, Maple's derivative (correct on the complex plane) agrees with the real-variable form found in the standard calculus text. For these five exceptions, this article explores two issues: (1) Does Maple's derivative, restricted to the real domain, agree with the real-variable form; and (2), to what extent do both forms agree on the complex plane.153726Wed, 10 Dec 2014 05:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Branch Cuts for a Product of Two Square-Roots
https://www.maplesoft.com/applications/view.aspx?SID=153697&ref=Feed
Naive simplification of f(z) = sqrt(z - 1) sqrt(z + 1) to F(z) = sqrt(z<sup>2</sup> - 1) results in a pair of functions that agree on only part of the complex plane. The enhanced ability of Maple 18 to find and display branch cuts of composite functions is used in this article to explore the branch cuts and regions of agreement/disagreement of f and F.<img src="/view.aspx?si=153697/thumb.jpg" alt="Classroom Tips and Techniques: Branch Cuts for a Product of Two Square-Roots" align="left"/>Naive simplification of f(z) = sqrt(z - 1) sqrt(z + 1) to F(z) = sqrt(z<sup>2</sup> - 1) results in a pair of functions that agree on only part of the complex plane. The enhanced ability of Maple 18 to find and display branch cuts of composite functions is used in this article to explore the branch cuts and regions of agreement/disagreement of f and F.153697Tue, 11 Nov 2014 05:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Locus of Eigenvalues
https://www.maplesoft.com/applications/view.aspx?SID=153463&ref=Feed
If P(s) is a parameter-dependent square matrix, what is the locus of its eigenvalues as s varies from, say, 0 to 1? For a non-square P, the eigenvalues can become complex, so the loci could exist as curves in the real or complex planes. To avoid these difficulties, consider only real symmetric matrices for which the loci of eigenvalues are real curves, but curves that could intersect. What does it mean to trace an individual eigenvalue of P(0) to P(1) if the eigenvalue has algebraic multiplicity more than 1?<img src="/view.aspx?si=153463/thumb.jpg" alt="Classroom Tips and Techniques: Locus of Eigenvalues" align="left"/>If P(s) is a parameter-dependent square matrix, what is the locus of its eigenvalues as s varies from, say, 0 to 1? For a non-square P, the eigenvalues can become complex, so the loci could exist as curves in the real or complex planes. To avoid these difficulties, consider only real symmetric matrices for which the loci of eigenvalues are real curves, but curves that could intersect. What does it mean to trace an individual eigenvalue of P(0) to P(1) if the eigenvalue has algebraic multiplicity more than 1?153463Fri, 15 Nov 2013 05:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Mathematical Thoughts on the Root Locus
https://www.maplesoft.com/applications/view.aspx?SID=153452&ref=Feed
Under suitable assumptions, the roots of the equation <em>f</em>(<em>z, c</em>) = 0, namely, <em>z</em> = <em>z</em>(<em>c</em>), trace a curve in the complex plane. In engineering feedback-control, such curves are called a <em>root locus</em>. This article examines the parameter-dependence of roots of polynomial and transcendental equations.<img src="/view.aspx?si=153452/thumb.jpg" alt="Classroom Tips and Techniques: Mathematical Thoughts on the Root Locus" align="left"/>Under suitable assumptions, the roots of the equation <em>f</em>(<em>z, c</em>) = 0, namely, <em>z</em> = <em>z</em>(<em>c</em>), trace a curve in the complex plane. In engineering feedback-control, such curves are called a <em>root locus</em>. This article examines the parameter-dependence of roots of polynomial and transcendental equations.153452Tue, 29 Oct 2013 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Slider-Control of Parameters in an ODE
https://www.maplesoft.com/applications/view.aspx?SID=152112&ref=Feed
Several ways to provide slider-control of parameters in a differential equation are considered. In particular, the cases of one and two parameters are illustrated, and for the case of two parameters, a 2-dimensional slider is constructed.<img src="/view.aspx?si=152112/thumb.jpg" alt="Classroom Tips and Techniques: Slider-Control of Parameters in an ODE" align="left"/>Several ways to provide slider-control of parameters in a differential equation are considered. In particular, the cases of one and two parameters are illustrated, and for the case of two parameters, a 2-dimensional slider is constructed.152112Mon, 23 Sep 2013 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Drawing a Normal and Tangent Plane on a Surface
https://www.maplesoft.com/applications/view.aspx?SID=150722&ref=Feed
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 LopezRate of Change of Surface Area on an Expanding Sphere
https://www.maplesoft.com/applications/view.aspx?SID=149511&ref=Feed
<p>An example of a related-rates problem in differential calculus that asks for the rate of change of surface area on a sphere whose volume expands at a constant rate is <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=7">solved here</a> via the syntax-free paradigm in Maple. </p>
<p>Recently, after presenting the solution in a Maplesoft Webinar, I was asked if it were possible to see an animation for this process. So, after a quick presentation of a solution, this worksheet will try to answer the request for an animation. Of course, we first have to consider just what is it that is to be displayed in the animation. It's easy enough to show an expanding sphere, but the question of real interest is the varying rate of change of surface area. How is the change in surface area to be visualized, let alone animated?</p><img src="/view.aspx?si=149511/related-rates.JPG" alt="Rate of Change of Surface Area on an Expanding Sphere" align="left"/><p>An example of a related-rates problem in differential calculus that asks for the rate of change of surface area on a sphere whose volume expands at a constant rate is <a href="http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=7">solved here</a> via the syntax-free paradigm in Maple. </p>
<p>Recently, after presenting the solution in a Maplesoft Webinar, I was asked if it were possible to see an animation for this process. So, after a quick presentation of a solution, this worksheet will try to answer the request for an animation. Of course, we first have to consider just what is it that is to be displayed in the animation. It's easy enough to show an expanding sphere, but the question of real interest is the varying rate of change of surface area. How is the change in surface area to be visualized, let alone animated?</p>149511Tue, 16 Jul 2013 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Solving Algebraic Equations by the Dragilev Method
https://www.maplesoft.com/applications/view.aspx?SID=149514&ref=Feed
The Dragilev method for solving certain systems of algebraic equations is used to parametrize the closed curve formed by the intersection of two given surfaces. This work is an elucidation of several posts to MaplePrimes.<img src="/view.aspx?si=149514/thumb.jpg" alt="Classroom Tips and Techniques: Solving Algebraic Equations by the Dragilev Method" align="left"/>The Dragilev method for solving certain systems of algebraic equations is used to parametrize the closed curve formed by the intersection of two given surfaces. This work is an elucidation of several posts to MaplePrimes.149514Tue, 16 Jul 2013 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: The Sliding Ladder
https://www.maplesoft.com/applications/view.aspx?SID=148714&ref=Feed
A January 10, 2013 post to <a href="http://www.mapleprimes.com/questions/142194-Sliding-Ladder-Animation" class="plainlink">MaplePrimes</a> asked for an animation of the trajectory traced by the center of a "sliding ladder." This month's article generalizes the solutions suggested by Adri van der Meer and Doug Meade, and shows the trajectory of an arbitrary point on the ladder as its top slides down a vertical wall and its bottom moves away from that wall along an orthogonal "floor." The location of the arbitrary point on the ladder is controlled by a slider, the animation being generated with the updated Explore command.<img src="/view.aspx?si=148714/thumb.jpg" alt="Classroom Tips and Techniques: The Sliding Ladder" align="left"/>A January 10, 2013 post to <a href="http://www.mapleprimes.com/questions/142194-Sliding-Ladder-Animation" class="plainlink">MaplePrimes</a> asked for an animation of the trajectory traced by the center of a "sliding ladder." This month's article generalizes the solutions suggested by Adri van der Meer and Doug Meade, and shows the trajectory of an arbitrary point on the ladder as its top slides down a vertical wall and its bottom moves away from that wall along an orthogonal "floor." The location of the arbitrary point on the ladder is controlled by a slider, the animation being generated with the updated Explore command.148714Fri, 21 Jun 2013 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Gems 31-35 from the Red Book of Maple Magic
https://www.maplesoft.com/applications/view.aspx?SID=147092&ref=Feed
Five additional "gems" from the Red Book of Maple Magic are detailed. Gem 31 shows how the updated Explore command can be applied to the numeric solution of an initial-value problem containing parameters. Gem 32 shows some list manipulations. Gem 33 clarifies some issues with the contourplot command, while Gem 34 clarifies some issues with the sample option in the plot command. Finally, Gem 36 examines the Equate command, and its alternatives.<img src="/view.aspx?si=147092/thumb.jpg" alt="Classroom Tips and Techniques: Gems 31-35 from the Red Book of Maple Magic" align="left"/>Five additional "gems" from the Red Book of Maple Magic are detailed. Gem 31 shows how the updated Explore command can be applied to the numeric solution of an initial-value problem containing parameters. Gem 32 shows some list manipulations. Gem 33 clarifies some issues with the contourplot command, while Gem 34 clarifies some issues with the sample option in the plot command. Finally, Gem 36 examines the Equate command, and its alternatives.147092Fri, 10 May 2013 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Bivariate Limits - Then and Now
https://www.maplesoft.com/applications/view.aspx?SID=145979&ref=Feed
An introductory overview of the functionalities in Maple's GraphTheory package.<img src="/view.aspx?si=145979/thumb.jpg" alt="Classroom Tips and Techniques: Bivariate Limits - Then and Now" align="left"/>An introductory overview of the functionalities in Maple's GraphTheory package.145979Wed, 17 Apr 2013 04:00:00 ZDr. Robert LopezDr. Robert Lopez