Calculus I: New Applications
http://www.maplesoft.com/applications/category.aspx?cid=175
en-us2017 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemThu, 24 Aug 2017 08:31:51 GMTThu, 24 Aug 2017 08:31:51 GMTNew applications in the Calculus I categoryhttp://www.mapleprimes.com/images/mapleapps.gifCalculus I: New Applications
http://www.maplesoft.com/applications/category.aspx?cid=175
Mathematics for Chemistry
https://www.maplesoft.com/applications/view.aspx?SID=154267&ref=Feed
This interactive electronic textbook in the form of Maple worksheets comprises two parts.
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Part I, mathematics for chemistry, is supposed to cover all mathematics that an instructor of chemistry might hope and expect that his students would learn, understand and be able to apply as a result of sufficient courses typically, but not exclusively, presented in departments of mathematics. Its nine chapters include (0) a summary and illustration of useful Maple commands, (1) arithmetic, algebra and elementary functions, (2) plotting, descriptive geometry, trigonometry, series, complex functions, (3) differential calculus of one variable, (4) integral calculus of one variable, (5) multivariate calculus, (6) linear algebra including matrix, vector, eigenvector, vector calculus, tensor, spreadsheet, (7) differential and integral equations, and (8) probability, distribution, treatment of laboratory data, linear and non-linear regression and optimization.
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Part II presents mathematical topics typically taught within chemistry courses, including (9) chemical equilibrium, (10) group theory, (11) graph theory, (12a) introduction to quantum mechanics and quantum chemistry, (14) applications of Fourier transforms in chemistry including electron diffraction, x-ray diffraction, microwave spectra, infrared and Raman spectra and nuclear-magnetic-resonance spectra, and (18) dielectric and magnetic properties of chemical matter.
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Other chapters are in preparation and will be released in due course.<img src="/view.aspx?si=154267/molecule.PNG" alt="Mathematics for Chemistry" align="left"/>This interactive electronic textbook in the form of Maple worksheets comprises two parts.
<BR><BR>
Part I, mathematics for chemistry, is supposed to cover all mathematics that an instructor of chemistry might hope and expect that his students would learn, understand and be able to apply as a result of sufficient courses typically, but not exclusively, presented in departments of mathematics. Its nine chapters include (0) a summary and illustration of useful Maple commands, (1) arithmetic, algebra and elementary functions, (2) plotting, descriptive geometry, trigonometry, series, complex functions, (3) differential calculus of one variable, (4) integral calculus of one variable, (5) multivariate calculus, (6) linear algebra including matrix, vector, eigenvector, vector calculus, tensor, spreadsheet, (7) differential and integral equations, and (8) probability, distribution, treatment of laboratory data, linear and non-linear regression and optimization.
<BR><BR>
Part II presents mathematical topics typically taught within chemistry courses, including (9) chemical equilibrium, (10) group theory, (11) graph theory, (12a) introduction to quantum mechanics and quantum chemistry, (14) applications of Fourier transforms in chemistry including electron diffraction, x-ray diffraction, microwave spectra, infrared and Raman spectra and nuclear-magnetic-resonance spectra, and (18) dielectric and magnetic properties of chemical matter.
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Other chapters are in preparation and will be released in due course.154267Tue, 30 May 2017 04:00:00 ZProf. John OgilvieProf. John OgilvieClassroom 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 LopezHollywood Math 2
https://www.maplesoft.com/applications/view.aspx?SID=153681&ref=Feed
<p>Over the years, Hollywood has entertained us with many mathematical moments in film and television, often in unexpected places. In this application, you’ll find several examples of Hollywood Math, including Fermat’s Last Theorem and <em>The Simpsons</em>, the Monty Hall problem in <em>21</em>, and a discussion of just how long that runway actually was in <em>The Fast and the Furious</em>. These examples are also presented in <a href="/webinars/recorded/featured.aspx?id=782">Hollywood Math 2: The Recorded Webinar</a>.</p>
<p>For even more examples, see <a href="/applications/view.aspx?SID=6611">Hollywood Math: The Original Episode</a>.</p><img src="/view.aspx?si=153681/HollywoodMath2.jpg" alt="Hollywood Math 2" align="left"/><p>Over the years, Hollywood has entertained us with many mathematical moments in film and television, often in unexpected places. In this application, you’ll find several examples of Hollywood Math, including Fermat’s Last Theorem and <em>The Simpsons</em>, the Monty Hall problem in <em>21</em>, and a discussion of just how long that runway actually was in <em>The Fast and the Furious</em>. These examples are also presented in <a href="/webinars/recorded/featured.aspx?id=782">Hollywood Math 2: The Recorded Webinar</a>.</p>
<p>For even more examples, see <a href="/applications/view.aspx?SID=6611">Hollywood Math: The Original Episode</a>.</p>153681Tue, 23 Sep 2014 04:00:00 ZMaplesoftMaplesoftRate 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: 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 LopezPerimeter, area and visualization of a plane figure
https://www.maplesoft.com/applications/view.aspx?SID=146470&ref=Feed
<p>The work contains three procedures that allow symbolically to calculate the perimeter and area of any plane figure bounded by <span>non-selfintersecting piecewise smooth curve</span>, and to portray this figure together with its boundary in a suitable design.</p><img src="/view.aspx?si=146470/planefigure_thumb.png" alt="Perimeter, area and visualization of a plane figure" align="left"/><p>The work contains three procedures that allow symbolically to calculate the perimeter and area of any plane figure bounded by <span>non-selfintersecting piecewise smooth curve</span>, and to portray this figure together with its boundary in a suitable design.</p>146470Tue, 30 Apr 2013 04:00:00 ZDr. Yury ZavarovskyDr. Yury ZavarovskyClassroom Tips and Techniques: Tractrix Questions - A Homework Problem from MaplePrimes
https://www.maplesoft.com/applications/view.aspx?SID=137299&ref=Feed
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 LopezMath Apps in Maple
https://www.maplesoft.com/applications/view.aspx?SID=132220&ref=Feed
Math Apps in Maple have give students and teachers the ability to explore and illustrate a wide variety of mathematical and scientific concepts. These fun and easy to use educational demonstrations are designed to illustrate various mathematical and physical concepts. This application contains a sampling of some of the many Math Apps available in Maple: drawing the graph of a quadratic, epicycloids, monte carlo approximations of pi, and throwing coconuts.<img src="/view.aspx?si=132220/mathapps_thumb.png" alt="Math Apps in Maple" align="left"/>Math Apps in Maple have give students and teachers the ability to explore and illustrate a wide variety of mathematical and scientific concepts. These fun and easy to use educational demonstrations are designed to illustrate various mathematical and physical concepts. This application contains a sampling of some of the many Math Apps available in Maple: drawing the graph of a quadratic, epicycloids, monte carlo approximations of pi, and throwing coconuts.132220Tue, 27 Mar 2012 04:00:00 ZMaplesoftMaplesoftzoMbi
https://www.maplesoft.com/applications/view.aspx?SID=129642&ref=Feed
<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 CyboulkoAn Epidemic Model (for Influenza or Zombies)
https://www.maplesoft.com/applications/view.aspx?SID=127836&ref=Feed
<p>Systems of differential equations can be used to model an epidemic of influenza or of zombies. This is an interactive Maple document suitable for use in courses on mathematical biology or differential equations or calculus courses that include differential equations. No knowledge of Maple is required.</p><img src="/view.aspx?si=127836/Cholera.jpg" alt="An Epidemic Model (for Influenza or Zombies)" align="left"/><p>Systems of differential equations can be used to model an epidemic of influenza or of zombies. This is an interactive Maple document suitable for use in courses on mathematical biology or differential equations or calculus courses that include differential equations. No knowledge of Maple is required.</p>127836Thu, 17 Nov 2011 05:00:00 ZDr. Robert IsraelDr. Robert IsraelWhy I Needed Maple to Make Cream Cheese Frosting
https://www.maplesoft.com/applications/view.aspx?SID=125069&ref=Feed
<p>A recipe for cream cheese frosting I was making called for 8 oz. (about 240 grams) of cream cheese. Unfortunately, I didn't have a kitchen scale, and the product I bought came in a 400 gram tub in the shape of a<strong> truncated cone</strong>, which has a rather cumbersome volume formula. <br />Given the geometry of this tub, how deep into the tub should I scoop to get 240 grams? The mathematics is trickier than you might think but lots of fun! And the final, tasty result is worth the effort!</p><img src="/view.aspx?si=125069/philly_thumb.png" alt="Why I Needed Maple to Make Cream Cheese Frosting" align="left"/><p>A recipe for cream cheese frosting I was making called for 8 oz. (about 240 grams) of cream cheese. Unfortunately, I didn't have a kitchen scale, and the product I bought came in a 400 gram tub in the shape of a<strong> truncated cone</strong>, which has a rather cumbersome volume formula. <br />Given the geometry of this tub, how deep into the tub should I scoop to get 240 grams? The mathematics is trickier than you might think but lots of fun! And the final, tasty result is worth the effort!</p>125069Tue, 23 Aug 2011 04:00:00 ZDr. Jason SchattmanDr. Jason SchattmanMapler. 05. Аlgebraic equations & Index
https://www.maplesoft.com/applications/view.aspx?SID=102285&ref=Feed
<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
https://www.maplesoft.com/applications/view.aspx?SID=102076&ref=Feed
<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 ZTIMOTIMOHow Fast Does An Advent Candle Burn?
https://www.maplesoft.com/applications/view.aspx?SID=100332&ref=Feed
<p>Any kid who's ever been entranced by an advent wreath knows that a tapered advent candle shrinks faster on Sunday night when it's new and slender than on Saturday night when it's old, stubby and caked with melted wax. How much faster? As an apropos application of math during this Christmas season, <strong>we derive a formula for the height of a burning tapered candle as a function of time</strong>. Assuming the candle has the shape of a cone when it is new and that it loses volume at a constant rate as it burns, we show that the height of the candle shrinks roughly in proportion to the cube root of time.</p><img src="/view.aspx?si=100332/thumb.jpg" alt="How Fast Does An Advent Candle Burn?" align="left"/><p>Any kid who's ever been entranced by an advent wreath knows that a tapered advent candle shrinks faster on Sunday night when it's new and slender than on Saturday night when it's old, stubby and caked with melted wax. How much faster? As an apropos application of math during this Christmas season, <strong>we derive a formula for the height of a burning tapered candle as a function of time</strong>. Assuming the candle has the shape of a cone when it is new and that it loses volume at a constant rate as it burns, we show that the height of the candle shrinks roughly in proportion to the cube root of time.</p>100332Mon, 20 Dec 2010 05:00:00 ZDr. Jason SchattmanDr. Jason SchattmanTwo Bodies Revolving Around Their Center of Mass with ANIMATION
https://www.maplesoft.com/applications/view.aspx?SID=99587&ref=Feed
<p>For any isolated system of two bodies revolving around each other by virtue of the gravitational attraction that each one exerts on the other, the general motion is best described by using a frame of reference attached to their common Center of Mass (CM). The reason is that their motion is in fact around their CM as we shall see. <br />For an isolated system the momentum remains constant so that the CM is either moving along a straight line or is at rest.<br />For an Earth's satellite we can always take the motion of the satellite relative to Earth using a geocentric frame of reference. <br />The reason is that:<br /> the mass of the satellite being insignificant compared to Earth's <br /> mass, the revolving satellite doesn't affect Earth at all so<br /> that the CM of Earth-satellite system is still the center of the Earth.<br /> Hence we use the center of the Earth as the origin of a rectangular<br /> coordinates system.<br /> <br />In this article we use Maple powerful animation routines to study the motion of two bodies having comparable masses revolving about each other by showing: <br />1- their combined motion as seen from their common Center of Mass,<br />2- their relative motion as if one of them is fixed and the other one is moving. <br />In this last instance the frame of reference is attached to the the body that is supposed to be at rest.<br /><br /></p><img src="/view.aspx?si=99587/thumb.jpg" alt="Two Bodies Revolving Around Their Center of Mass with ANIMATION" align="left"/><p>For any isolated system of two bodies revolving around each other by virtue of the gravitational attraction that each one exerts on the other, the general motion is best described by using a frame of reference attached to their common Center of Mass (CM). The reason is that their motion is in fact around their CM as we shall see. <br />For an isolated system the momentum remains constant so that the CM is either moving along a straight line or is at rest.<br />For an Earth's satellite we can always take the motion of the satellite relative to Earth using a geocentric frame of reference. <br />The reason is that:<br /> the mass of the satellite being insignificant compared to Earth's <br /> mass, the revolving satellite doesn't affect Earth at all so<br /> that the CM of Earth-satellite system is still the center of the Earth.<br /> Hence we use the center of the Earth as the origin of a rectangular<br /> coordinates system.<br /> <br />In this article we use Maple powerful animation routines to study the motion of two bodies having comparable masses revolving about each other by showing: <br />1- their combined motion as seen from their common Center of Mass,<br />2- their relative motion as if one of them is fixed and the other one is moving. <br />In this last instance the frame of reference is attached to the the body that is supposed to be at rest.<br /><br /></p>99587Mon, 29 Nov 2010 05:00:00 ZDr. Ahmed BaroudyDr. Ahmed BaroudyClassroom Tips and Techniques: Fitting Circles in Space to 3-D Data
https://www.maplesoft.com/applications/view.aspx?SID=1644&ref=Feed
<p>In "A Project on Circles in Space," Carl Cowen provided an algebraic solution for the problem of fitting a circle to a set of points in space. His technique used the singular value decomposition from linear algebra, and was recast as a project in the volume ATLAST: Computer Exercises for Linear Algebra. Both versions of the problem used MATLAB® for the calculations. In this worksheet, we implement the algebraic calculations in Maple, then add noise to the data to test the robustness of the algebraic method. Next, we solve the problem with an analytic approach that incorporates least squares, and appears to be more robust in the face of noisy data. Finally, the analytic approach leads to explicit formulas for the fitting circle, so we end with graphs of the data, fitting circle, and plane lying closest to the data in the least-squares sense.</p>
<p><em><sub>Simulink is a registered trademark of The MathWorks, Inc.</sub></em></p><img src="/view.aspx?si=1644/thumb3.jpg" alt="Classroom Tips and Techniques: Fitting Circles in Space to 3-D Data" align="left"/><p>In "A Project on Circles in Space," Carl Cowen provided an algebraic solution for the problem of fitting a circle to a set of points in space. His technique used the singular value decomposition from linear algebra, and was recast as a project in the volume ATLAST: Computer Exercises for Linear Algebra. Both versions of the problem used MATLAB® for the calculations. In this worksheet, we implement the algebraic calculations in Maple, then add noise to the data to test the robustness of the algebraic method. Next, we solve the problem with an analytic approach that incorporates least squares, and appears to be more robust in the face of noisy data. Finally, the analytic approach leads to explicit formulas for the fitting circle, so we end with graphs of the data, fitting circle, and plane lying closest to the data in the least-squares sense.</p>
<p><em><sub>Simulink is a registered trademark of The MathWorks, Inc.</sub></em></p>1644Mon, 17 May 2010 04:00:00 ZDr. Robert LopezDr. Robert LopezTeach the intuitive limit with Maple without learning any Maple commands
https://www.maplesoft.com/applications/view.aspx?SID=35364&ref=Feed
<p><span lang="EN-CA">For each 2-hour calculus course, there are two worksheets for the students. First, there’s the Theory Worksheet, which allow students to learn concepts as well as methods. These worksheets can be used in a classroom where all the students have a computer, and the teacher projects his worksheet on the screen. In this way, all the students go through the worksheets at the same time with the teacher.</span></p>
<p><span lang="EN-CA"><span lang="EN-CA">The second worksheet is the Exercise Worksheet which students complete individually. There are mandatory exercises for the students to do and once these are completed, there are randomly generated Additional Exercises if the student needs more practice. Furthermore, the student is graded at the end of each exercise set, allowing him/her to gauge his/her progress. None of these worksheets requires knowing any Maple commands.</span></span></p>
<p><span lang="EN-CA"><span lang="EN-CA"><span lang="EN-CA">As an example, I have provided both worksheets for the introductory course on limits. The notion is presented intuitively graphically as well as numerically. In addition, the students are also taught how to read and write limits using mathematical notation and they have exercises to complete in the second worksheet.</span></span></span></p>
<p><span lang="EN-CA"><span lang="EN-CA"><span lang="EN-CA"><span lang="EN-CA">These worksheets are prepared and ready for use by any teacher. Therefore, the teacher still has the advantage of teaching mathematics with Maple, such as randomly generating exercises, plotting graphs, without having to program or to learn any Maple commands.</span></span></span></span></p>
<p><span lang="EN-CA"><span lang="EN-CA"><span lang="EN-CA"><span lang="EN-CA"><span>Vous trouverez deux feuilles Maple interactives que l’on peut utiliser sans connaître aucune commande Maple. La feuille théorique permet d’apprendre la notion intuitive de limite.<span> </span>Elle peut être utilisée en classe avec les étudiants qui ont chacun un ordinateur. <span> </span>La feuille d’exercices permet de faire des exercices. Certains exercices sont obligatoires et d’autres sont générés aléatoirement. L’étudiant reçoit aussi une note.</span></span></span></span></span></p><img src="/view.aspx?si=35364/intuitive_limit_ver.jpg" alt="Teach the intuitive limit with Maple without learning any Maple commands" align="left"/><p><span lang="EN-CA">For each 2-hour calculus course, there are two worksheets for the students. First, there’s the Theory Worksheet, which allow students to learn concepts as well as methods. These worksheets can be used in a classroom where all the students have a computer, and the teacher projects his worksheet on the screen. In this way, all the students go through the worksheets at the same time with the teacher.</span></p>
<p><span lang="EN-CA"><span lang="EN-CA">The second worksheet is the Exercise Worksheet which students complete individually. There are mandatory exercises for the students to do and once these are completed, there are randomly generated Additional Exercises if the student needs more practice. Furthermore, the student is graded at the end of each exercise set, allowing him/her to gauge his/her progress. None of these worksheets requires knowing any Maple commands.</span></span></p>
<p><span lang="EN-CA"><span lang="EN-CA"><span lang="EN-CA">As an example, I have provided both worksheets for the introductory course on limits. The notion is presented intuitively graphically as well as numerically. In addition, the students are also taught how to read and write limits using mathematical notation and they have exercises to complete in the second worksheet.</span></span></span></p>
<p><span lang="EN-CA"><span lang="EN-CA"><span lang="EN-CA"><span lang="EN-CA">These worksheets are prepared and ready for use by any teacher. Therefore, the teacher still has the advantage of teaching mathematics with Maple, such as randomly generating exercises, plotting graphs, without having to program or to learn any Maple commands.</span></span></span></span></p>
<p><span lang="EN-CA"><span lang="EN-CA"><span lang="EN-CA"><span lang="EN-CA"><span>Vous trouverez deux feuilles Maple interactives que l’on peut utiliser sans connaître aucune commande Maple. La feuille théorique permet d’apprendre la notion intuitive de limite.<span> </span>Elle peut être utilisée en classe avec les étudiants qui ont chacun un ordinateur. <span> </span>La feuille d’exercices permet de faire des exercices. Certains exercices sont obligatoires et d’autres sont générés aléatoirement. L’étudiant reçoit aussi une note.</span></span></span></span></span></p>35364Tue, 13 Apr 2010 04:00:00 ZDr. Jean-Philippe VilleneuveDr. Jean-Philippe VilleneuveClassroom Tips and Techniques: Visualizing Regions of Integration
https://www.maplesoft.com/applications/view.aspx?SID=34062&ref=Feed
<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 Lopez