In this post, I would like to share some exercises that I recently taught to an undergraduate student using Maple. These exercises aimed to deepen their understanding of mathematical concepts through computational exploration and visualization. With its powerful symbolic computation capabilities, Maple proved to be an excellent tool for this purpose. Below, I present a few of the exercises and the insights they provided. Interestingly, the student found Maple to be more user-friendly and efficient compared to the software he usually uses for his studies. Below, I present a few of the exercises and the insights they provided.

One of the first topics we tackled was the Fourier series. We used Maple to illustrate how the Fourier series approximates a given function as more terms are added. We explored this through both static plots and interactive animations.

To help the student understand the behavior of different types of functions, we defined piecewise functions using Maple's piecewise command. This allowed us to model functions that behave differently over various intervals, such as the following cubic function exercise

Maple's Explore command was an effective tool for creating an interactive learning environment. We used it to create sliders that allowed the student to vary parameters, such as the number of terms in a Fourier series, and see the immediate impact on the plot.

Download Fourier_Series.mw

The first time they are evaluated, some aborts can occur; the second time they are evaluated, no exception is thrown: 
 

Download run__twice.mw

Why does (the outer) RootOf have to be evaluated twice? 
Note. There are other similar examples, but they are less concise (so they are omitted here).

If you draw a chord in any curve, when the latter becomes infinitely small, the ratio of the surface segment to the triangle formed by the chord and the associated tangents is 2:3.

(Source: Archive of Mathematics and Physics, editor Johann August Grunert, 31st part of 1858, pp. 449-453, "On a remarkable general theorem on curves",
Author: Andreas Völler)

(The curve may be assumed to be sufficiently differentiable.)

I have a quite complex expression (where and are real numbers): 

According to coulditbe, is satisfiable: 

_EnvTry := 'hard':
coulditbe(expr) assuming real;
 = 
                              true

But according to SMTLIB:-Satisfiable, is not satisfiable: 

SMTLIB:-Satisfiable(expr) assuming real;
 = 
                             false

Why are the two results opposite? 

For reference, below is the output from RealDomain:-solve: 

RealDomain:-solve(expr);
 = 
               /           1  2\                 
              { p = p, q = - p  }, {p = p, q = 0}
               \           4   /                 

I also tried using RealDomain:-simplify, yet the output remains almost unchanged (Why?). 

Bernoulli first order ode has form as show in wikipedia  and also on Maple own site as

Notice that it is P(x)*y above and not P(x)* y^(-1) so the y(x) must be linear in that term.   But when I give Maple this ode

ode:=diff(y(x),x) + x*y(x)^(-1)= y(x)^(-1);

Which is clearly not of the form above, it solves it as Bernoulli.  In the above ode, P(x) is x and Q(x) is 1 and n is -1.

The ode advisor correctly said it is separable. But trace shows it used Bernoulli. Also when asking it to solve it as Bernoulli, it does.

What Am I missing here?  Is it not wrong for Maple to use Bernoulli method on this ode which is not Bernoulli?

Worksheet below

Download why_this_ode_bernullli_sept_15_2024.mw

Dear Maple community,

I was going to export my math assignment about vector functions as an PDF, when I noticed that it butchered every column vector printed out in blue in the document as math-output. I only have this issue when exporting my work.

How do I make Maple display both coordinates of vector in the blue output field?

I produced the below image as an illustration of my problem:

Thank you in advance.

A classic result states that the equation x³＋px²＋qx＋r＝0 with real coefficients p, q, r has positive roots iff p＜0, q＞0, r＜0 and -27r² - 2p(2p² - 9q)r + q²(p² - 4q) ⩾ 0 (see for example this question). 
However, Maple appears unable to find the condition: 

a, b, c := allvalues(RootOf(x^3 + p*x^2 + q*x + r, x), 'implicit'):
RealDomain:-solve({a, b, c} >~ 0, [p, q, r]);
 = 
Warning, solutions may have been lost
                               []

Is there a way to get the above conditions in Maple with as little human intervention as possible (I mean, without a priori knowledge of the theory of polynomials)? 

Edit. An interesting problem is when these three positive roots can further be the lengths of sides of a triangle. For reference, here are some (unenlightening) results from some other software: 

I'm trying to solve a system of coupled differential equations numerically, but I'm getting the following error

Error, (in dsolve/numeric/process_input) system must be entered as a set/list of expressions/equations

The error occurs at the dsolve step, despite trying to ensure that all equations and conditions are in the correct form (sets/lists).

Could someone help me identify what I'm missing here?

Thanks in advance!

Download dsolve_error.mw

I wanted to change  eq:= 1/2 * sqrt(-2*lambda)  to 1/2 %* sqrt(-2*lambda)  using a rule.

It works outside of rule ofcourse. But when I put %* in the RHS of the rule, maple hangs. It seems it is going into infinite loop.

I tried the trick of using '%*' but this gives syntax error.

Same problem happens when using %. and not just %*

Is there a workaround?

Attached worksheet. Make sure to save all work before trying it.
 

Download maples_hangs_applyrule_sept_10_2024.mw

Hello!

I present a simple work-up of rolling a plane curve along a fixed plane curve in 2d space. Maple sources are attached.

Kind regards!

Source.zip

Hello,

Please help me solve this type of equations, I have more of them, but there is only one in the attached file. fsolve does not work. Please help.

Best regards!

tg_Test.mw

The help page of interface('longdelim') states: 

If I set , the Maple Input

try f:=0 catch:end try;

can be converted into 

via , but why is

use f=0 in f+1 end use;

still converted into

 instead of “use f = 0 in f + 1 end”? 

With the attached task I would like to learn how Maple handles polynomials and plots. For this I have chosen a task with an interesting history from 1593. Adriaan van Roomen posed it to F. Vieta, who solved it after a short consideration.

My questions are:
1.) How are very long terms entered and displayed in Maple in an appropriate manner?
2.) How are the graphs of several functions displayed in the same coordinate system?
3.) Which graphics settings must be selected for differentiable functions in order to obtain nice, rounded curves rather than angular ones from the numerical result?
4.) Can the apparently random oscillations of the curve at the end of the interval be suppressed?
5.) The curves for p(x) and sinnx(x) are theoretically identical, since p(x) is the trigonometric expansion of sinnx(x). Their graphs must therefore be identical. How can this be displayed?
6.) The polynomial has many real zeros. How can the zeros be clearly presented in a table?

AF_20240909.mw

This is an excersise in one of Mathematica's tutorials. The problem is to find from all persumtations of [1,2,3,4] those lists which has 2 in them before 3.

But the idea is that 2 can be anywhere before the 3, with possibly zero or more numbers between.

Will show how to do this in that other software, and ask if there is a way using Maple pattern matching or better way to do this in Maple better than what I did.

L=Permutations[{1,2,3,4}]
Cases[L,{___,2,___,3,___}]

gives

{{1, 2, 3, 4}, {1, 2, 4, 3}, {1, 4, 2, 3}, {2, 1, 3, 4}, 
{2, 1, 4,  3}, {2, 3, 1, 4}, {2, 3, 4, 1}, {2, 4, 1, 3}, 
{2, 4, 3, 1}, {4, 1,  2, 3}, {4, 2, 1, 3}, {4, 2, 3, 1}}

We see in all of the above, 2 is before 3.

In that other software, the ___ means there is zero or more things.  I could not find how to do this in Maple's pattern matching. So had to use has and then find the index of 2 and 3 in each list and check if the index of 2 is smaller than the index of 3. Which is kinda awakard and not as elegent as using a pattern, but it gives same result.

L:=combinat:-permute([1,2,3,4]);
map(X->`if`(has(X,2) and has(X,3) and ListTools:-Search(2,X)<ListTools:-Search(3,X),X,NULL) ,L);

Gives

[[1, 2, 3, 4], [1, 2, 4, 3], [1, 4, 2, 3], [2, 1, 3, 4], 
[2, 1, 4, 3], [2, 3, 1, 4], [2, 3, 4, 1], [2, 4, 1, 3], 
[2, 4, 3, 1], [4, 1, 2, 3], [4, 2, 1, 3], [4, 2, 3, 1]]

Can you suggest a way using either patmatch or applyrules in Maple to do the same?  

Could this be done easier than what I did using evalindents without the need for pattern?

I find patterns easier to work with myself.

ps. converting each list to string, and then using regular expression or string matching, is not what I am looking for. 

ps. the check in Maple code of has(X,2) and has(X,3) is not really needed here, since we know each list will have 2 and 3 in them. But I kept these to make it more general for other type of problems where these extra checks might be needed.

I am trying to create the plot for the dynamics of the discrete Klein - the equation gives Gordon equation (DKG) with friction for t=0 and t=3000.

As you can see below, creating the plot for t=0 is pretty easy. However, for t=3000, I tried to create a procedure to produce the desired plots, but when I ran the code, nothing happened. The worksheets are evaluated all the time. There are no results.

My target is the following plots

Download Numerical_Simulation_of_discrete_Klein-Gordon_Equation.mw