MaplePrimes Posts

MaplePrimes Posts are for sharing your experiences, techniques and opinions about Maple, MapleSim and related products, as well as general interests in math and computing.

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  • A new Maple e-book, Multivariate Calculus Study Guide, is now available. Part of the Clickable Calculus collection of interactive Maple e-books, this guide takes full advantage of Maple’s Clickable Math approach. It has over 600 worked examples, the vast majority of which are solved using interactive, Clickable Math techniques. 

    Deisgned to help students taking this course, instructors may also find this e-book useful as a guide to using Clickable Math to teach Multivariate Calculus.

    See Multivariate Calculus Study Guide for more information.



    In addition to the Maple 2015.2 and MapleSim 2015.2 updates for Mac, we have just released updates to both Maple and MapleSim for Windows and Linux.

    Maple 2015.2a provides a fix for the sum bug reported here.

    MapleSim 2015.a provides a variety of interface improvements, and updates to the MapleSim Battery Library and MapleSim CAD Toolbox.

    For Mac users, these improvements are included with the Maple 2015.2/MapleSim 2015.2 updates.

    All updates are available through the Check for Updates system, and are also available from our website on the downloads section of our website.


    We have just released updates to Maple 2015 and MapleSim 2015 that fix the problems on Mac OS X 10.11 (El Capitan).  If you want to use the new OS, you should update your products.

    Updates are available through Check for Updates and from the Downloads section of our website. See Maple 2015.2 and MapleSim 2015.2 for details. MapleSim users, please note that this update also gives you all the new features in MapleSim 2015.2.

    If you are using earlier versions of these products, please read the  Maple and MapleSim on Mac OS X 10.11 FAQ for more information about your options.



    Consider the well-known Euler's formula  

     eix = cos x + i sin x   

    When we calculate that for  x = π  we get:

    eiπ = cos π + i sin π   or

    eiπ = −1 + i × 0   (because cos π = −1 and sin π = 0)  or

    eiπ = −1  or  eiπ + 1 = 0

    It seems absolutely magical that such a neat equation combines  5  fundamental constants: e ,  i ,  π , 1 , 0

    The purpose of this post - to give a simple visualization of equality  eiπ = −1  (statical and animated) by expanding  eiπ  in a series of complex numbers. These numbers we represent as vectors in the plane. We will see that the partial sums of this series are broken lines like a spiral, twisting around the point -1 steadily approaching to it.

    Euler procedure has one required parameter  n is positive integer - the number of displayed terms of the series  for  eiπ  

    Optional parameter  a  is any symbol (by default  a=NULL). We use this option if  instead of a static spiral want to see an animated spiral. 

    Procedure code can be found in the attached file


    Examples of use.

    The first example shows  8 terms of the series (broken line of 8 units):





    The terms of the series where  n> = 10  on the same plot can not be seen as very small. In this case, we use  the second plot with magnification of  100 : 1 .  

    The second example:





    In the third example, we see an animated broken line. It's  first 9 units represented  on the left plot, and then for n> = 10 on the right plot:

    Euler(13, a);



    This January 28th, we will be hosting another full-production, live streaming webinar featuring an all-star cast of Maplesoft employees: Andrew Rourke (Director of Teaching Solutions), Jonny Zivku (Maple T.A. Product Manager), and Daniel Skoog (Maple Product Manager). Attend the webinar to learn how educators all around the world are using Maple and Maple T.A. in their own classrooms.

    Any STEM educator, administrator, or curriculum coordinator who is interested in learning how Maple and Maple T.A. can help improve student grades, reduce drop-out rates, and save money on administration costs will benefit from attending this webinar.

    Click here for more information and registration.

    In this paper we will demonstrate the importance of using simple to complex algorithms applied to complex systems in civil and mechanical engineering. In order to develop solutions that developers need to be involved in issues of advanced dynamic computer science. We show how is that with the Maple scientific program and through component-based algorithms can generate power then then be inserted into specific algorithms. Will form patterns with movements of rotation and revolution of their axes, in each case to model and analyze the curves thereof comprising. With these modelalos and curve analysis we can predict manufacturing costs, freight, inter alia estrcturas which they can be used with the correct use of Maplesoft.



    (in spanish)

    Lenin Araujo Castillo





    The Joint Mathematics Meetings are taking place this week (January 6 – 9) in Seattle, Washington, U.S.A. This will be the 99th annual winter meeting of the Mathematical Association of America (MAA) and the 122nd annual meeting of the American Mathematical Society (AMS).

    Maplesoft will be exhibiting at booth #203 as well as in the networking area. Please stop by our booth or the networking area to chat with me and other members of the Maplesoft team, as well as to pick up some free Maplesoft swag or win some prizes.

    Given the size of the Joint Math Meetings, it can be challenging to pick which events to attend. Hopefully we can help by suggesting a few Maple-related talks and events:

    Maplesoft is hosting a catered reception and presentation ‘Challenges of Modern Education: Bringing Math Instruction Online’ on Thursday, January 7th at 18:00 in the Cedar Room at the Seattle Sheraton. You can find more details and registration information here:

    Another not to miss Maple event is “30 Years of Digitizing Mathematical Knowledge with Maple”, presented by Edgardo Cheb-Terrab, on Thursday, January 7 at 10:00 in Room 603 of the Convention Center.

    Here’s a list of Maple-related events and talks:

    Exploration of Mathematics Teaching and Assessment through Maple-Software Projects of Art Diagram Design as Undergraduate Student Research Projects

    Wednesday, Jan 6, 10:20, Room 2B, Convention Center

    Lina Wu


    30 Years of Digitizing Mathematical Knowledge with Maple

    Thursday, Jan 7, 10:00, Room 603, Convention Center

    Edgardo Cheb-Terrab


    MAA Poster Session – Collaborative Research: Maplets for Calculus

    Thursday, Jan 7, 14:00, Hall 4F, 4th Floor, Convention Center


    Challenges of Modern Education: Bringing Math Instruction Online

    Thursday, Jan 7, 18:00, Cedar Room, 2nd Floor, Sheraton Center


    Using Maple to Promote Modelling in Differential Equations

    Friday, Jan 8, 10:40, Room 617, Convention Center

    Patrice G Tiffany; Rosemary C Farley


    If you are presenting at Joint Math and would like to advertise your Maple-related talk, please feel free to comment below, or send me a message with your event and I’ll add it to the list above.


    See you in Seattle!


    Maple Product Manager

    This post is my attempt to answer the question from here .  

    The procedure  ContoursWithLabels  has 2 required parameters: Expr  is an expression in  x  and  y  variables,  Range1  and  Range2  are ranges for  x  and  y . In this case, the output is the list of floats for the contours and 8 black contours (with labels) (the axis of coordinates as a box). 

    The optional parameters: Number is positive integer - the number of contours (by default Number=8),  S is a set of real numbers  C  for contours (for which Expr=C) (by default  S={}),  GraphicOptions  is a list of graphic options for plotting (by default  GraphicOptions=[color = black, axes = box]),  Coloring  is an equality  Coloring=list of color options for  plots[dencityplot]  command (by default Coloring=NULL). 

    The code of the procedure:


    ContoursWithLabels := proc (Expr, Range1::(range(realcons)), Range2::(range(realcons)), Number::posint := 8, S::(set(realcons)) := {}, GraphicOptions::list := [color = black, axes = box], Coloring::`=` := NULL)

    local r1, r2, L, f, L1, h, S1, P, P1, r, M, C, T, p, p1, m, n, A, B, E;

    uses plots, plottools;

    f := unapply(Expr, x, y);

    if S = {} then r1 := rand(convert(Range1, float)); r2 := rand(convert(Range2, float));

    L := [seq([r1(), r2()], i = 1 .. 205)];

    L1 := convert(sort(select(a->type(a, realcons), [seq(f(op(t)), t = L)]), (a, b) ->is(abs(a) < abs(b))), set);

    h := (L1[-6]-L1[1])/Number;

    S1 := [seq(L1[1]+(1/2)*h+h*(n-1), n = 1 .. Number)] else

    S1 := convert(S, list)  fi;

    print(Contours = evalf[2](S1));

    r := k->rand(20 .. k-20); M := []; T := [];

    for C in S1 do

    P := implicitplot(Expr = C, x = Range1, y = Range2, op(GraphicOptions), gridrefine = 3);

    P1 := [getdata(P)];

    for p in P1 do

    p1 := convert(p[3], listlist); n := nops(p1);

    if n < 500 then m := `if`(40 < n, (r(n))(), round((1/2)*n)); M := `if`(40 < n, [op(M), p1[1 .. m-11], p1[m+11 .. n]], [op(M), p1]); T := [op(T), [op(p1[m]), evalf[2](C)]] else

    if 500 <= n then h := floor((1/2)*n); m := (r(h))(); M := [op(M), p1[1 .. m-11], p1[m+11 .. m+h-11], p1[m+h+11 .. n]]; T := [op(T), [op(p1[m]), evalf[2](C)], [op(p1[m+h]), evalf[2](C)]]

    fi; fi; od; od;

    A := plot(M, op(GraphicOptions));

    B := plots:-textplot(T);

    if Coloring = NULL then E := NULL else E := ([plots:-densityplot])(Expr, x = Range1, y = Range2, op(rhs(Coloring)))  fi;

    display(E, A, B);

    end proc:


    Examples of use:

    ContoursWithLabels(x^2+y^2, -3 .. 3, -3 .. 3);




    ContoursWithLabels(x^2-y^2, -5 .. 5, -5 .. 5, {-20, -15, -10, -5, 0, 5, 10, 15, 20}, [color = black, thickness = 2, axes = box], Coloring = [colorstyle = HUE, colorscheme = ["White", "Red"], style = surface]);




    The next example, I took from here:

    ContoursWithLabels(sin(1.3*x)*cos(.9*y)+cos(.8*x)*sin(1.9*y)+cos(.2*x*y), -5 .. 0, 2 .. 5, {seq(-2 .. 2, 0.5)}, [color = black, axes = box], Coloring = [colorstyle = HUE, colorscheme = ["Cyan", "Red"], style = surface]);



    There are many more examples can be found in the attached file.


    Edit. The attached file has been corrected.

    The year 2015 has been one with interesting and relevant developments in the MathematicalFunctions  and FunctionAdvisor projects.


    Gaps were filled regarding mathematical formulas, with more identities for all of BesselI, BesselK, BesselY, ChebyshevT, ChebyshevU, Chi, Ci, FresnelC, FresnelS, GAMMA(z), HankelH1, HankelH2, InverseJacobiAM, the twelve InverseJacobiPQ for P, Q in [C,D,N,S], KelvinBei, KelvinBer, KelvinKei, KelvinKer, LerchPhi, arcsin, arcsinh, arctan, ln;


    Developments happened in the Mathematical function package, to both compute with symbolic sequences and symbolic nth order derivatives of algebraic expressions and functions;


    The input FunctionAdvisor(differentiate_rule, mathematical_function) now returns both the first derivative (old behavior) and the nth symbolic derivative (new behavior) of a mathematical function;


    A new topic, plot, used as FunctionAdvisor(plot, mathematical_function), now returns 2D and 3D plots for each mathematical function, following the NIST Digital Library of Mathematical Functions;


    The previously existing FunctionAdvisor(display, mathematical_function) got redesigned, so that it now displays more information about any mathematical function, and organized into a Section with subsections for each of the different topics, making it simpler to find the information one needs without getting distracted by a myriad of formulas that are not related to what one is looking for.

    More mathematics


    More mathematical knowledge is in place, more identities, differentiation rules of special functions with respect to their parameters, differentiation of functions whose arguments involve symbolic sequences with an indeterminate number of operands, and sum representations for special functions under different conditions on the functions' parameters.



    More powerful symbolic differentiation (nth order derivative)


    Significative developments happened in the computation of the nth order derivative of mathematical functions and algebraic expressions involving them.



    Mathematical handling of symbolic sequences


    Symbolic sequences enter various formulations in mathematics. Their computerized mathematical handling, however, was never implemented - only a representation for them existed in the Maple system. In connection with this, a new subpackage, Sequences , within the MathematicalFunctions package, has been developed.



    Visualization of mathematical functions


    When working with mathematical functions, it is frequently desired to have a rapid glimpse of the shape of the function for some sampled values of their parameters. Following the NIST Digital Library of Mathematical Functions, a new option, plot, has now been implemented.



    Section and subsections displaying properties of mathematical functions


    Until recently, the display of a whole set of mathematical information regarding a function was somehow cumbersome, appearing all together on the screen. That display was and is still available via entering, for instance for the sin function, FunctionAdvisor(sin) . That returns a table of information that can be used programmatically.

    With time however, the FunctionAdvisor evolved into a consultation tool, where a better organization of the information being displayed is required, making it simpler to find the information we need without being distracted by a screen full of complicated formulas.

    To address this requirement, the FunctionAdvisor now returns the information organized into a Section with subsections, built using the DocumentTools package. This enhances the presentation significantly.



    These developments can be installed in Maple 2015 as usual, by downloading the updates (bundled with the Physics and Differential Equations updates) from the Maplesoft R&D webpage for Mathematical Functions and Differential Equations


    Edgardo S. Cheb-Terrab
    Physics, Differential Equations and Mathematical Functions, Maplesoft

    Just a simple little proc that grabs and displays some simple data for general stars (I've only listed a couple for starters).  It wasn't till later that I realized it's not an up to date database as it doesn't contain more recent interesting stars like Kepler 452.  One would have to go to the SIMBAD astronomical database for which I have not yet devoted any time for .. at least not yet. 

    StarData := proc (a::string) local b, c, d, e1, e2, e3; b := StringTools:-DeleteSpace(StringTools:-Substitute(a, " ", "+")); c := HTTP:-Get(cat("", b)); d := convert(c[2], string); e1 := d[StringTools:-Search("<H1>", d)+4 .. StringTools:-Search("</H1>", d)-1]; e2 := d[StringTools:-Search("Right", d) .. StringTools:-Search("Standard error in distance", d)-8]; e3 := StringTools:-SubstituteAll(StringTools:-SubstituteAll(StringTools:-SubstituteAll(e2, "<BR>", ""), "<B>", ""), "</B>", ""); print(e1); print(e3) end proc:


    "Right Ascension and Declination: 0h0m0s, +0&deg;0'0" (epoch 2000.0)
Distance from Sol: 0 light-years (0 parsecs)


    StarData("Wolf 1061")

    "Right Ascension and Declination: 16h30m18.097s, -12&deg;39'45.17" (epoch 2000.0)
Distance from Sol: 13.91 light-years (4.264 parsecs)


    StarData("Alpha Centauri")

    "Right Ascension and Declination: 14h39m35.88s, -60&deg;50'7.4" (epoch 2000.0)
Distance from Sol: 4.395 light-years (1.347 parsecs)



    "Right Ascension and Declination: 6h45m8.871s, -16&deg;42'57.99" (epoch 2000.0)
Distance from Sol: 8.601 light-years (2.637 parsecs)








    I'd like to pay attention to an article J, B. van den Berg and J.-P. Lessard, Notices of the AMS, October 2015, p. 1057-1063.  We know numerous  applications of CASes to algebra. The authors present such  applications to dynamics. It would be interesting and useful to obtain  opinions of Maple experts on this topic.

    Here is its introduction:

    "Nonlinear dynamics shape the world around us, from the harmonious movements of celestial bod-
    ies,  via  the  swirling  motions  in  fluid  flows,  to the  complicated  biochemistry  in  the  living  cell.
    Mathematically  these  beautiful  phenomena  are modeled by nonlinear dynamical systems, mainly
    in  the  form  of  ordinary  differential  equations (ODEs), partial differential equations (PDEs) and
    delay differential equations (DDEs). The presence of nonlinearities severely complicates the mathe-
    matical analysis of these dynamical systems, and the difficulties are even greater for PDEs and DDEs,
    which are naturally defined on infinite-dimensional function spaces. With the availability of powerful
    computers and sophisticated software, numerical simulations have quickly become the primary tool
    to study the models. However, while the pace of progress increases, one may ask: just how reliable
    are our computations? Even for finite-dimensional ODEs, this question naturally arises if the system
    under  study  is  chaotic,  as  small  differences  in initial conditions (such as those due to rounding
    errors  in  numerical  computations)  yield  wildly diverging outcomes. These issues have motivated
    the development of the field of rigorous numerics in dynamics"

    As the year draws to a close, we start looking forward to a new year and a new release of Maple. With every new release comes many new features and updates to explore.

    We are looking for several new beta testers with a good working knowledge of Maple; We need your input, your ideas, and your experience with our products to help us improve the software and get it ready for general release.

    There are many benefits to becoming a beta tester:

    • You’ll get to use the new software before anyone else does.
    • You’ll help us make our software better in ways that work for you.
    • Your suggestions could determine the future direction of the software.
    • You’ll get feedback right from the development team.

    If you are interested in becoming a beta tester for the next version of Maple, please email: beta (at) for more information.

    The well known William Lowell Putnam Mathematical Competition (76th edition)  took place this month.
    Here is a Maple approach for two of the problems.

    1. For each real number x, 0 <= x < 1, let f(x) be the sum of  1/2^n  where n runs through all positive integers for which floor(n*x) is even.
    Find the infimum of  f.
    (Putnam 2015, A4 problem)

    local n, s:=0;
    for n to N do
      if type(floor(n*x),even) then s:=s+2^(-n) fi;
      #if floor(n*x) mod 2 = 0  then s:=s+2^(-n) fi;

    plot(f, 0..0.9999);


    min([seq(f(t), t=0.. 0.998,0.0001)]);



    So, the infimum is 4/7.
    Of course, this is not a rigorous solution, even if the result is correct. But it is a valuable hint.
    I am not sure if in the near future, a CAS will be able to provide acceptable solutions for such problems.

    2. If the function f  is three times differentiable and  has at least five distinct real zeros,
    then f + 6f' + 12f'' + 8f''' has at least two distinct real zeros.
    (Putnam 2015, B1 problem)

    F := f + 6*D(f) + 12*(D@@2)(f) + 8*(D@@3)(f);


    We are sugested to consider

    g3:=diff(g, x$3);


    So, F(x) = k(x) * g3 = k(x) * g'''
    g  has 5 distinct zeros implies g''' and hence F have 5-3=2 distinct zeros, q.e.d.


    A modified version of John May's modification of Bruce Char's animated Christmas tree, using some Maple 2015 features to automatically scale the 3D plot and automatically play the animation.

    I went with the viewpoint option, and removed the blinking lights. In the worksheet the tree shows as scaled larger. Perhaps it could be a submission to walkingrandomly.

    The CIELAB perceptual model of human vision can be used to predict the color of a wavelength in sRGB. I found the CIEDE2000 corrections to hue, chroma, and lighness were required to get the right values in the blue to violet region. The CIEDE2000 results seem pretty close to what I see looking through a diffraction grating considering I have no way of knowing how my eyes are adapted.

    Use the new, faster  version. It can run in under 1 minute;   and  WtoC_data.xlsx

    You will need the data for the color matching functions in the WtoC Excel File. The data directory can now be controlled in the mw file.


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