Dr. Robert Lopez: New Applications
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en-us2015 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemSun, 01 Feb 2015 22:51:48 GMTSun, 01 Feb 2015 22:51:48 GMTNew applications published by Dr. Robert Lopezhttp://www.mapleprimes.com/images/mapleapps.gifDr. Robert Lopez: New Applications
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Classroom Tips and Techniques: Real and Complex Derivatives of Some Elementary Functions
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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
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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
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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
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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
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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
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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 LopezRate of Change of Surface Area on an Expanding Sphere
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<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
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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
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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
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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
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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 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 LopezGems 26-30 from the Red Book of Maple Magic
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<p>In 2011, this column published five "Maple Magic" articles, each containing five "gems" gleaned from interactions with Maple and the Maplesoft programmers. Here are five more recent additions to the Red Book, every one of which contained something about Maple that was a surprise to me, experienced Maple user that I am.</p><img src="/view.aspx?si=141091/thumb.jpg" alt="Gems 26-30 from the Red Book of Maple Magic" align="left"/><p>In 2011, this column published five "Maple Magic" articles, each containing five "gems" gleaned from interactions with Maple and the Maplesoft programmers. Here are five more recent additions to the Red Book, every one of which contained something about Maple that was a surprise to me, experienced Maple user that I am.</p>141091Tue, 04 Dec 2012 05:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Least-Squares Fits
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<p><span id="ctl00_mainContent__documentViewer" ><span ><span class="body summary">The least-squares fitting of functions to data can be done in Maple with eleven different commands from four different packages. The <em>CurveFitting</em> and LinearAlgebra packages each have a LeastSquares command; the Optimization package has the LSSolve and NLPSolve commands; and the Statistics package has the seven commands Fit, LinearFit, PolynomialFit, ExponentialFit, LogarithmicFit, PowerFit, and NonlinearFit, which can return some measure of regression analysis.</span></span></span></p><img src="/view.aspx?si=140942/image.jpg" alt="Classroom Tips and Techniques: Least-Squares Fits" align="left"/><p><span id="ctl00_mainContent__documentViewer" ><span ><span class="body summary">The least-squares fitting of functions to data can be done in Maple with eleven different commands from four different packages. The <em>CurveFitting</em> and LinearAlgebra packages each have a LeastSquares command; the Optimization package has the LSSolve and NLPSolve commands; and the Statistics package has the seven commands Fit, LinearFit, PolynomialFit, ExponentialFit, LogarithmicFit, PowerFit, and NonlinearFit, which can return some measure of regression analysis.</span></span></span></p>140942Wed, 28 Nov 2012 05:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Applying the Epsilon-Delta Definition of a Limit
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This month's article looks at how the "epsilon-delta" definition of a limit can be implemented in Maple: Take δ(ε) as the smaller of δ<sub><em>L</em></sub> and δ<sub><em>R</em></sub>, themselves determined by the equations ƒ(<em>a</em> + δ<sub><em>R</em></sub>) = <em>L</em> + ε and ƒ(<em>a</em> - δ<sub><em>L</em></sub>) = <em>L</em> - ε, and then show |ƒ(<em>a</em> + <em>t</em> δ) - <em>L</em>| < ε, where 0 < |<em>t</em>| < 1.<img src="/view.aspx?si=140239/thumb.jpg" alt="Classroom Tips and Techniques: Applying the Epsilon-Delta Definition of a Limit" align="left"/>This month's article looks at how the "epsilon-delta" definition of a limit can be implemented in Maple: Take δ(ε) as the smaller of δ<sub><em>L</em></sub> and δ<sub><em>R</em></sub>, themselves determined by the equations ƒ(<em>a</em> + δ<sub><em>R</em></sub>) = <em>L</em> + ε and ƒ(<em>a</em> - δ<sub><em>L</em></sub>) = <em>L</em> - ε, and then show |ƒ(<em>a</em> + <em>t</em> δ) - <em>L</em>| < ε, where 0 < |<em>t</em>| < 1.140239Mon, 12 Nov 2012 05:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Tractrix Questions - A Homework Problem from MaplePrimes
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A May 13, 2012, post to MaplePrimes asked some interesting questions about the tractrix defined parametrically by <em>x(s)</em> = sech<em>(s), y(s)</em> = <em>s</em> - tanh(s), s ≥ 0. I answered these questions on May 14 in a worksheet that forms the basis for this month's article.</p>
<p>It behooves me to write this article because the solution given in the May 14 MaplePrimes reply wasn't completely correct, the error stemming from a confounding of the variables <em>x, y, </em>and <em>s</em>. Mea culpa.<img src="/view.aspx?si=137299/thumb.jpg" alt="Classroom Tips and Techniques: Tractrix Questions - A Homework Problem from MaplePrimes" align="left"/>A May 13, 2012, post to MaplePrimes asked some interesting questions about the tractrix defined parametrically by <em>x(s)</em> = sech<em>(s), y(s)</em> = <em>s</em> - tanh(s), s ≥ 0. I answered these questions on May 14 in a worksheet that forms the basis for this month's article.</p>
<p>It behooves me to write this article because the solution given in the May 14 MaplePrimes reply wasn't completely correct, the error stemming from a confounding of the variables <em>x, y, </em>and <em>s</em>. Mea culpa.137299Wed, 12 Sep 2012 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Best Taylor-Polynomial Approximations
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In the early 90s, Joe Ecker (Rensselaer Polytechnic Institute) provided a Maple solution to the problem of determining for a given function, which expansion point in a specified interval yielded the best quadratic Taylor polynomial approximation, where "best" was measured by the L<sub>2</sub>-norm. This article applies Ecker's approach to the function <em>f(x)</em> = sinh<em>(x)</em> – <em>x e<sub>-3x</sub>,</em> -1 ≤ <em>x</em> ≤ 3, then goes on to find other approximating quadratic polynomials.<img src="/view.aspx?si=136471/image.jpg" alt="Classroom Tips and Techniques: Best Taylor-Polynomial Approximations" align="left"/>In the early 90s, Joe Ecker (Rensselaer Polytechnic Institute) provided a Maple solution to the problem of determining for a given function, which expansion point in a specified interval yielded the best quadratic Taylor polynomial approximation, where "best" was measured by the L<sub>2</sub>-norm. This article applies Ecker's approach to the function <em>f(x)</em> = sinh<em>(x)</em> – <em>x e<sub>-3x</sub>,</em> -1 ≤ <em>x</em> ≤ 3, then goes on to find other approximating quadratic polynomials.136471Tue, 14 Aug 2012 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: Slider-Control of Parameters in Numeric Solutions of ODEs
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In the article "Sliders for Parameter-Dependent Curves", and again in the article "Caustics for a Plane Curve", the use of sliders to control parameters was explored. This month's article explores the use of sliders to control parameters in a differential equation that must be solved numerically.<img src="/view.aspx?si=135062/thumb.jpg" alt="Classroom Tips and Techniques: Slider-Control of Parameters in Numeric Solutions of ODEs" align="left"/>In the article "Sliders for Parameter-Dependent Curves", and again in the article "Caustics for a Plane Curve", the use of sliders to control parameters was explored. This month's article explores the use of sliders to control parameters in a differential equation that must be solved numerically.135062Tue, 12 Jun 2012 04:00:00 ZDr. Robert LopezDr. Robert Lopez