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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  • There seems to be a bug with improper integration:

    integrate(cos(t)*exp(-x*t),t=-infinity..infinity)

    gives

    0

    Substituting any number for x, or assuming x >= 0  (or x<=0) does give the correct result,

    The problem also persists when assuming x>-1 (or x>-Maple_floats(MIN_FLOAT))

    Hi MaplePrimes,

    another_recursive_sequence.mw

    another_recursive_sequence.pdf

    These two files have the same content.  One is a .pdf and the other is a Maple Worksheet.  I explore integer sequences of the form - 

    a(r) = c*a(r-1)+d*a(r-2) with a(1) and a(2) given.

    Some of these sequences are in (the Online Encyclopedia of Integer Sequences) OEIS.org and some are not.  If we restrict c to 1 and assume that a(1)=1 and a(2) = 2 we have the parameter d remaining.  See additional webpage - 

    https://sites.google.com/site/recrusivefunction/

    Let me know if you like the code.

    Regards,

    Matt

     

     

    Books free. Like!!!

    Lenin Araujo Castillo

    As you can see this app performs the trace of a given path r (t), then locate the position vector in a specific time. It also graphs the velocity vector, acceleration, Tangential and Normal unit vectors, along with the Binormal. Very good app developed entirely in Maple for our engineering students.

    Plot_of_Position_Vector_UPDATED.mw

    https://youtu.be/OzAwShHHXq8

    Lenin Araujo Castillo

    Ambassador of Maple

    It seems a large number of people, when initially using maple, wrongly deduce that for example sin(60) is the sin of 60 degrees and not the sin of 1/3 Pi radians.  I believe mathematica's default is degrees.  When a student compares an expression to another but forgets to realize a value is read as radians and not degrees they are perplexed when Maple returns false and Mathematica returns true.

    As a suggestion, under tools->options allow a user to be able to change how maple reads values within trigonometric funtions as either radians or degrees.

    Most times when someone computes the sin(60) what they really mean in Maple..
    is sin(convert(60 degrees, radians))

     

    I'm back from presenting work in the "23rd Conference on Applications of Computer Algebra -2017" . It was a very interesting event. This fifth presentation, about "The Appell doubly hypergeometric functions", describes a very recent project I've been working at Maple, i.e. the very first complete computational implementation of the Appell doubly hypergeometric functions. This work appeared in Maple 2017. These functions have a tremendous potential in that, at the same time, they have a myriad of properties, and include as particular cases most of the existing mathematical language, and so they have obvious applications in integration, differential equations, and applied mathematics all around. I think these will be the functions of this XXI century, analogously to what happened with hypergeometric functions in the previous century.

    At the end, there is a link to the presentation worksheet, with which one could open the sections and reproduce the presentation examples.
     

    The four double-hypergeometric Appell functions,

    a complete implementation in a computer algebra system

     

    Edgardo S. Cheb-Terrab

    Physics, Differential Equations and Mathematical Functions, Maplesoft

     

    Abstract:
    The four multi-parameter Appell functions, AppellF1 , AppellF2 , AppellF3  and AppellF4  are doubly hypergeometric functions that include as particular cases the 2F1 hypergeometric  and some cases of the MeijerG  function, and with them most of the known functions of mathematical physics. Appell functions have been popping up with increasing frequency in applications in quantum mechanics, molecular physics, and general relativity. In this talk, a full implementation of these functions in the Maple computer algebra system, including, for the first time, their numerical evaluation over the whole complex plane, is presented, with details about the symbolic and numerical strategies used.

    Appell Functions (symbolic)

     

     

    The main references:

    • 

    P. Appel, J.Kamke de Feriet, "Fonctions hypergeometriques et Hyperspheriques", 1926

    • 

    H. Srivastava, P.W. Karlsson, "Multiple Gaussian Hypergeometric Series", 1985

    • 

    24 papers in the literature, ranging from 1882 to 2015

     

    Definition and Symmetries

       

    Polynomial and Singular Cases

       

    Single Power Series with Hypergeometric Coefficients

       

    Analytic Extension from the Appell Series to the Appell Functions

       

    Euler-Type and Contiguity Identities

       

    Appell Differential Equations

       

    Putting all together

       

    Problem: some formulas in the literature are wrong or miss the conditions indicating when are they valid (exchange with the Mathematics director of the DLMF - NIST)

       

    Appell Functions (numeric)

     

     

    Goals

     

    • 

    Compute these Appell functions over the whole complex plane

    • 

    Considering that this is a research problem, implement different methods and flexible optional arguments to allow for:

    a) comparison between methods (both performance and correctness),

    b) investigation of a single method in different circumstances.

    • 

    Develop a computational structure that can be reused with other special functions (abstract code and provide the main options), and that could also be translated to C (so: only one numerical implementation, not 100 special function numerical implementations)

    Limitation: the Maple original evalf command does not accept optional arguments

     

    The cost of numerically evaluating an Appell function

     

    • 

    If it is a special hypergeometric case, then between 1 to 2 hypergeometric functions

    • 

    Next simplest case (series/recurrence below) 3 to 4 hypergeometric functions plus adding somewhat large formulas that involve only arithmetic operations up to 20,000 times (frequently less than 100 times)

    • 

    Next simplest case: the formulas themselves are power series with hypergeometric function coefficients; these cases frequently converge rapidly but may involve the numerical evaluation of up to hundreds of hypergeometric functions to get the value of a single Appell function.

     

    Strategy for the numerical evaluation of Appell functions (or other functions ...)

     

     

    The numerical evaluation flows orderly according to:

    1) check whether it is a singular case

    2) check whether it is a special value

    3) compute the value using a series derived from a recurrence related to the underlying ODE

    4) perform an sum using an infinite sum formula, checking for convergence

    5) perform the numerical integration of the ODE underlying the given Appell function

    6) perform a sequence of concatenated Taylor series expansions

    Examples

       

    Series/recurrence

       

    Numerical integration of an underlying differential equation (ODEs and dsolve/numeric)

       

    Concatenated Taylor series expansions covering the whole complex plane

       

    Subproducts

     

    Improvements in the numerical evaluation of hypergeometric functions

       

    Evalf: an organized structure to implement the numerical evaluation of special functions in general

       

    To be done

       


     

    Download Appell_Functions.mw   
    Download Appell_Functions.pdf

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

    I'm back from presenting work in the "23rd Conference on Applications of Computer Algebra -2017" . It was a very interesting event. This fourth presentation, about "The FunctionAdvisor: extending information on mathematical functions with computer algebra algorithms", describes the FunctionAdvisor project at Maple, a project I started working during 1998, where the key idea I am trying to explore is that we do not need to collect a gazillion of formulas but just core blocks of mathematical information surrounded by clouds of algorithms able to derive extended information from them. In this sense this is also unique piece of software: it can derive properties for rather general algebraic expressions, not just well known tabulated functions. The examples illustrate the idea.

    At the end, there is a link to the presentation worksheet, with which one could open the sections and reproduce the presentation examples.
     

    The FunctionAdvisor: extending information on mathematical functions

    with computer algebra algorithms

     

    Edgardo S. Cheb-Terrab

    Physics, Differential Equations and Mathematical Functions, Maplesoft

     

    Abstract:

    A shift in paradigm is happening, from: encoding information into a database, to: encoding essential blocks of information together with algorithms within a computer algebra system. Then, the information is not only searchable but can also be recreated in many different ways and actually used to compute. This talk focuses on this shift in paradigm over a real case example: the digitizing of information regarding mathematical functions as the FunctionAdvisor project of the Maple computer algebra system.

    The FunctionAdvisor (basic)

       

    Beyond the concept of a database

     
      

    " Mathematical functions, are defined by algebraic expressions. So consider algebraic expressions in general ..."

    Formal power series for algebraic expressions

       

    Differential polynomial forms for algebraic expressions

       

    Branch cuts for algebraic expressions

       

    The nth derivative problem for algebraic expressions

       

    Conversion network for mathematical and algebraic expressions

       

    References

       


     

    Download FunctionAdvisor.mw

    Download FunctionAdvisor.pdf

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

    I'm back from presenting work in the "23rd Conference on Applications of Computer Algebra -2017" . It was a very interesting event. This third presentation, about "Computer Algebra in Theoretical Physics", describes the Physics project at Maplesoft, also my first research project at University, that evolved into the now well-known Maple Physics package. This is a unique piece of software and perhaps the project I most enjoy working.

    At the end, there is a link to the presentation worksheet, with which one could open the sections and reproduce the presentation examples.
     

     

     

    Computer Algebra in Theoretical Physics

     

    Edgardo S. Cheb-Terrab

    Physics, Differential Equations and Mathematical Functions, Maplesoft

     

    Abstract:

     

    Generally speaking, physicists still experience that computing with paper and pencil is in most cases simpler than computing on a Computer Algebra worksheet. On the other hand, recent developments in the Maple system have implemented most of the mathematical objects and mathematics used in theoretical physics computations, and have dramatically approximated the notation used in the computer to the one used with paper and pencil, diminishing the learning gap and computer-syntax distraction to a strict minimum.

     

    In this talk, the Physics project at Maplesoft is presented and the resulting Physics package is illustrated by tackling problems in classical and quantum mechanics, using tensor and Dirac's Bra-Ket notation, general relativity, including the equivalence problem, and classical field theory, deriving field equations using variational principles.

     

     

     

     

    ... and why computer algebra?

     

    We can concentrate more on the ideas instead of on the algebraic manipulations

     

    We can extend results with ease

     

    We can explore the mathematics surrounding a problem

     

    We can share results in a reproducible way

     

    Representation issues that were preventing the use of computer algebra in Physics

       

    Classical Mechanics

     

    *Inertia tensor for a triatomic molecule

       

    Quantum mechanics

     

    *The quantum operator components of  `#mover(mi("L",mathcolor = "olive"),mo("&rarr;",fontstyle = "italic"))` satisfy "[L[j],L[k]][-]=i `&epsilon;`[j,k,m] L[m]"

       

    *Unitary Operators in Quantum Mechanics

     

    *Eigenvalues of an unitary operator and exponential of Hermitian operators

       

    *Properties of unitary operators

     

     

    Consider two set of kets " | a[n] >" and "| b[n] >", each of them constituting a complete orthonormal basis of the same space.

    *Verify that "U=(&sum;) | b[k] >< a[k] |" , maps one basis to the other, i.e.: "| b[n] >=U | a[n] >"

       

    *Show that "U=(&sum;) | b[k] > < a[k] | "is unitary

       

    *Show that the matrix elements of U in the "| a[n] >" and  "| b[n] >" basis are equal

       

    Show that A and `&Ascr;` = U*A*`#msup(mi("U"),mo("&dagger;"))`have the same spectrum (eigenvalues)

       

    Schrödinger equation and unitary transform

       

    Translation operators using Dirac notation

       

    *Quantization of the energy of a particle in a magnetic field

       

    Classical Field Theory

     

    The field equations for the lambda*Phi^4 model

       

    *Maxwell equations departing from the 4-dimensional Action for Electrodynamics

       

    *The Gross-Pitaevskii field equations for a quantum system of identical particles

       

    General Relativity

     

    Exact Solutions to Einstein's Equations  Lambda*g[mu, nu]+G[mu, nu] = 8*Pi*T[mu, nu]

       

    *"Physical Review D" 87, 044053 (2013)

       

    The Equivalence problem between two metrics

       

    *On the 3+1 split of the 4D Einstein equations

       

    Tetrads and Weyl scalars in canonical form

       

     

     


     

    Download Physics.mw

    Download Physics.pdf

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

    I'm back from presenting work in the "23rd Conference on Applications of Computer Algebra - 2017" . It was a very interesting event. This second presentation, about "Differential algebra with mathematical functions, symbolic powers and anticommutative variables", describes a project I started working in 1997 and that is at the root of Maple's dsolve and pdsolve performance with systems of equations. It is a unique approach. Not yet emulated in any other computer algebra system.

    At the end, there is a link to the presentation worksheet, with which one could open the sections and reproduce the presentation examples.
     

    Differential algebra with mathematical functions,

    symbolic powers and anticommutative variables

     

    Edgardo S. Cheb-Terrab

    Physics, Differential Equations and Mathematical Functions, Maplesoft

     

    Abstract:
    Computer algebra implementations of Differential Algebra typically require that the systems of equations to be tackled be rational in the independent and dependent variables and their partial derivatives, and of course that A*B = A*B, everything is commutative.

     

    It is possible, however, to extend this computational domain and apply Differential Algebra techniques to systems of equations that involve arbitrary compositions of mathematical functions (elementary or special), fractional and symbolic powers, as well as anticommutative variables and functions. This is the subject of this presentation, with examples of the implementation of these ideas in the Maple computer algebra system and its ODE and PDE solvers.

     

     

    restartwith(PDEtools); interface(imaginaryunit = i)

    sys := [diff(xi(x, y), y, y) = 0, -6*(diff(xi(x, y), y))*y+diff(eta(x, y), y, y)-2*(diff(xi(x, y), x, y)) = 0, -12*(diff(xi(x, y), y))*a^2*y-9*(diff(xi(x, y), y))*a*y^2-3*(diff(xi(x, y), y))*b-3*(diff(xi(x, y), x))*y-3*eta(x, y)+2*(diff(eta(x, y), x, y))-(diff(xi(x, y), x, x)) = 0, -8*(diff(xi(x, y), x))*a^2*y-6*(diff(xi(x, y), x))*a*y^2+4*(diff(eta(x, y), y))*a^2*y+3*(diff(eta(x, y), y))*a*y^2-4*eta(x, y)*a^2-6*eta(x, y)*a*y-2*(diff(xi(x, y), x))*b+(diff(eta(x, y), y))*b-3*(diff(eta(x, y), x))*y+diff(eta(x, y), x, x) = 0]

     

    declare((xi, eta)(x, y))

    xi(x, y)*`will now be displayed as`*xi

     

    eta(x, y)*`will now be displayed as`*eta

    (1)

    for eq in sys do eq end do

    diff(diff(xi(x, y), y), y) = 0

     

    -6*(diff(xi(x, y), y))*y+diff(diff(eta(x, y), y), y)-2*(diff(diff(xi(x, y), x), y)) = 0

     

    -12*(diff(xi(x, y), y))*a^2*y-9*(diff(xi(x, y), y))*a*y^2-3*(diff(xi(x, y), y))*b-3*(diff(xi(x, y), x))*y-3*eta(x, y)+2*(diff(diff(eta(x, y), x), y))-(diff(diff(xi(x, y), x), x)) = 0

     

    -8*(diff(xi(x, y), x))*a^2*y-6*(diff(xi(x, y), x))*a*y^2+4*(diff(eta(x, y), y))*a^2*y+3*(diff(eta(x, y), y))*a*y^2-4*eta(x, y)*a^2-6*eta(x, y)*a*y-2*(diff(xi(x, y), x))*b+(diff(eta(x, y), y))*b-3*(diff(eta(x, y), x))*y+diff(diff(eta(x, y), x), x) = 0

    (2)

    casesplit(sys)

    `casesplit/ans`([eta(x, y) = 0, diff(xi(x, y), x) = 0, diff(xi(x, y), y) = 0], [])

    (3)

    NULL

    Differential polynomial forms for mathematical functions (basic)

       

    Differential polynomial forms for compositions of mathematical functions

       

    Generalization to many variables

       

    Arbitrary functions of algebraic expressions

       

    Examples of the use of this extension to include mathematical functions

       

    Differential Algebra with anticommutative variables

       


     

    Download DifferentialAlgebra.mw

    Download DifferentialAlgebra.pdf

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

     

    I'm back from presenting work in the "23rd Conference on Applications of Computer Algebra - 2017" . It was a very interesting event. This first presentation, about "Active Learning in High-School Mathematics using Interactive Interfaces", describes a project I started working 23 years ago, which I believe will be part of the future in one or another form. This is work actually not related to my work at Maplesoft.

    At the end, there is a link to the presentation worksheet, with which one could open the sections and reproduce the presentation examples.
     

     

    Active learning in High-School mathematics using Interactive Interfaces

     

    Edgardo S. Cheb-Terrab

    Physics, Differential Equations and Mathematical Functions, Maplesoft

     

    Abstract:


    The key idea in this project is to learn through exploration using a web of user-friendly Highly Interactive Graphical Interfaces (HIGI). The HIGIs, structured as trees of interlinked windows, present concepts using a minimal amount of text while maximizing the possibility of visual and analytic exploration. These interfaces run computer algebra software in the background. Assessment tools are integrated into the learning experience within the general conceptual map, the Navigator. This Navigator offers students self-assessment tools and full access to the logical sequencing of course concepts, helping them to identify any gaps in their knowledge and to launch the corresponding learning interfaces. An interactive online set of HIGIS of this kind can be used at school, at home, in distance education, and both individually and in a group.

     

     

    Computer algebra interfaces for High-School students of "Colegio de Aplicação"  (UERJ/1994)

       

    Motivation

     

     

    When we are the average high-school student facing mathematics, we tend to feel

     

    • 

    Bored, fragmentarily taking notes, listening to a teacher for 50 or more minutes

    • 

    Anguished because we do not understand some math topics (too many gaps accumulated)

    • 

    Powerless because we don't know what to do to understand (don't have any instant-tutor to ask questions and without being judged for having accumulated gaps)

    • 

    Stressed by the upcoming exams where the lack of understanding may become evident

     

    Computer algebra environments can help in addressing these issues.

     

     

    • 

    Be as active as it can get while learning at our own pace.

    • 

    Explore at high speed and without feeling judged. There is space for curiosity with no computational cost.

    • 

    Feel empowered by success. That leads to understanding.

    • 

    Possibility for making of learning a social experience.

     

    Interactive interfaces

     

     

     

    Interactive interfaces do not replace the teacher - human learning is an emotional process. A good teacher leading good active learning is a positive experience a student will never forget

     

     

    Not every computer interface is a valuable resource, at all. It is the set of pedagogical ideas implemented that makes an interface valuable (the same happens with textbooks)

     

     

    A course on high school mathematics using interactive interfaces - the Edukanet project

     

     

    – 

    Brazilian and Canadian students/programmers were invited to participate - 7 people worked in the project.

     

    – 

    Some funding provided by the Brazilian Research agency CNPq.

    Tasks:

    -Develop a framework to develop the interfaces covering the last 3 years of high school mathematics (following the main math textbook used in public schools in Brazil)

    - Design documents for the interfaces according to given pedagogical guidelines.

    - Create prototypes of Interactive interfaces, running Maple on background, according to design document and specified layout (allow for everybody's input/changes).

     

    The pedagogical guidelines for interactive interfaces

       

    The Math-contents design documents for each chapter

     

    Example: complex numbers

       

    Each math topic:  a interactive interrelated interfaces (windows)

     

     

    For each topic of high-school mathematics (chapter of a textbook), develop a tree of interactive interfaces (applets) related to the topic (main) and subtopics

     

    Example: Functions

     

    • 

    Main window

     

    • 

    Analysis window

    • 

     

    • 

    Parity window

    • 

    Visualization of function's parity

    • 

    Step-by-Step solution window

    The Navigator: a window with a tile per math topic

     

     

     

    • 

    Click the topic-tile to launch a smaller window, topic-specific, map of interrelated sub-topic tiles, that indicates the logical sequence for the sub-topics, and from where one could launch the corresponding sub-topic interactive interface.

    • 

    This topic-specific smaller window allows for identifying the pre-requisites and gaps in understanding, launching the corresponding interfaces to fill the gaps, and tracking the level of familiarity with a topic.

     

     

     

     

     

    The framework to create the interfaces: a version of NetBeans on steroids ...

       

    Complementary classroom activity on a computer algebra worksheet

     

     

    This course is organized as a guided experience, 2 hours per day during five days, on learning the basics of the Maple language, and on using it to formulate algebraic computations we do with paper and pencil in high school and 1st year of undergraduate science courses.

     

    Explore. Having success doesn't matter, using your curiosity as a compass does - things can be done in so many different ways. Have full permission to fail. Share your insights. All questions are valid even if to the side. Computer algebra can transform the learning of mathematics into interesting understanding, success and fun.

    1. Arithmetic operations and elementary functions

       

    2. Algebraic Expressions, Equations and Functions

       

    3. Limits, Derivatives, Sums, Products, Integrals, Differential Equations

       

    4. Algebraic manipulation: simplify, factorize, expand

       

    5. Matrices (Linear Algebra)

       

     

    Advanced students: guiding them to program mathematical concepts on a computer algebra worksheet

       

    Status of the project

     

     

    Prototypes of interfaces built cover:

     

    • 

    Natural numbers

    • 

    Functions

    • 

    Integer numbers

    • 

    Rational numbers

    • 

    Absolute value

    • 

    Logarithms

    • 

    Numerical sequences

    • 

    Trigonometry

    • 

    Matrices

    • 

    Determinants

    • 

    Linear systems

    • 

    Limits

    • 

    Derivatives

    • 

    Derivative of the inverse function

    • 

    The point in Cartesian coordinates

    • 

    The line

    • 

    The circle

    • 

    The ellipse

    • 

    The parabole

    • 

    The hyperbole

    • 

    The conics

    More recent computer algebra frameworks: Maple Mobius for online courses and automated evaluation

       

     


     

    Download Computer_Algebra_in_Education.mw

    Download Computer_Algebra_in_Education.pdf

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

    We have just released an update to Maple. Maple 2017.2 includes updated translations for Japanese, Traditional Chinese, Simplified Chinese, Brazilian Portuguese, French, and Spanish. It also contains improvements to the MapleCloud, physics, limits, and PDEs. This update is available through Tools>Check for Updates in Maple, and is also available from our website on the Maple 2017.2 download page.

     Eithne


     

    A geometric construction for the Summer Holiday

     
    Does every plane simple closed curve contain all four vertices of some square?

     This is an old classical conjecture. See:
    https://en.wikipedia.org/wiki/Inscribed_square_problem

    Maybe someone finds a counterexample (for non-analytic curves) using the next procedure and becomes famous!

     

    SQ:=proc(X::procedure, Y::procedure, rng::range(realcons), r:=0.49)
    local t1:=lhs(rng), t2:=rhs(rng), a,b,c,d,s;
    s:=fsolve({ X(a)+X(c) = X(b)+X(d),
                Y(a)+Y(c) = Y(b)+Y(d),
                (X(a)-X(c))^2+(Y(a)-Y(c))^2 = (X(b)-X(d))^2+(Y(b)-Y(d))^2,
                (X(a)-X(c))*(X(b)-X(d)) + (Y(a)-Y(c))*(Y(b)-Y(d)) = 0},
              {a=t1..t1+r*(t2-t1),b=rng,c=rng,d=t2-r*(t2-t1)..t2});  #lprint(s);
    if type(s,set) then s:=rhs~(s)[];[s,s[1]] else WARNING("No solution found"); {} fi;
    end:

     

    Example

     

    X := t->(10-sin(7*t)*exp(-t))*cos(t);
    Y := t->(10+sin(6*t))*sin(t);
    rng := 0..2*Pi;

    proc (t) options operator, arrow; (10-sin(7*t)*exp(-t))*cos(t) end proc

     

    proc (t) options operator, arrow; (10+sin(6*t))*sin(t) end proc

     

    0 .. 2*Pi

    (1)

    s:=SQ(X, Y, rng):
    plots:-display(
       plot([X,Y,rng], scaling=constrained),
       plot([seq( eval([X(t),Y(t)],t=u),u=s)], color=blue, thickness=2));

     

    It appears google doesn't know about the haversine formula.  Huh?  Well at least google can't draw the proper path for it.  I typed in google "distance from Pyongyang to NewYork city"  and got 10,916km.  Ok that's fine but then it drew a map

    The map path definitely did not look right.  Pulled out my globe traced a rough path of the one google showed and I got 13 inches (where 1 inch=660miles) -> 8580 miles = 13808 km .. clearly looks like google goofed. 

    So we need Maple to show us the proper path.
     

    with(DataSets):
    with(Builtin):
    m := WorldMap();
    AddPath(m, [-74.0059, 40.7128], [125.7625, 39.0392]):
    Display(m):
    

    Ok so you say that really doesn't look like the shortest path.  Well, lets visualize that on the globe projection

    Display(m, projection = Globe, orientation = [-180, 0, 0])

    Ah, now it is clear
    Pyonyang_to_NewYork.mw

     

    As a momentary diversion, I threw together a package that downloads map images into Maple using the Google Static Maps API.

    If you have Maple 2017, you can install the package using the MapleCloud Package Manager or by executing PackageTools:-Install("5769608062566400").

    This worksheet has several examples, but I thought I'd share a few below .

    Here's the Maplesoft office

     

    Let's view a roadmap of Waterloo, Ontario.

     

    The package features over 80 styles for roadmaps. These are examples of two styles (the second is inspired by the art of Piet Mondrian and the De Stijl movement)

     

    You can also find the longitude and latitude of a location (courtesy of Google's Geocoding API). Maple returns a nested list if it finds multiple locations.

     

    The geocoding feature can also be used to add points to Maple 2017's built-in world maps.

     

    Let me know what you think!

    The representation of the tangent plane in the form of a square with a given length of the side at any point on the surface.

    The equation of the tangent plane to the surface at a given point is obtained from the condition that the tangent plane is perpendicular to the normal vector. With the aid of any auxiliary point not lying on this normal to the surface, we define the direction on the tangent plane. From the given point in this direction, we lay off segments equal to half the length of the side of our square and with the help of these segments we construct the square itself, lying on the tangent plane with the center at a given point.

    An examples of constructing tangent planes at points of the same intersection line for two surfaces.
    Tangent_plane.mw

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