Consider the well-known Euler's formula  

 e^ix = cos x + i sin x   

When we calculate that for  x = π  we get:

e^iπ = cos π + i sin π   or

e^iπ = −1 + i × 0   (because cos π = −1 and sin π = 0)  or

e^iπ = −1  or  e^iπ + 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  e^iπ = −1  (statical and animated) by expanding  e^iπ  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  e^iπ  

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  Euler.mw

Examples of use.

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

Euler(8);

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:

Euler(14);

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);

Euler.mw

I want to have a matrix such that 

Crosslinking surfaces by a spiral.



My fantasy is a source.

eq1 := z = y*log(x): eq2 := z = y+x*log(x):

DispIntersecting := implicitplot3d([eq1, eq2], x = 0 .. 10, y = -30 .. 30, z = -40 .. 40, color = [blue, green]):

solve({eq1, eq2}, [x, y, z]);

assign(%):

DispIntersection := spacecurve([x, y, z], x = 0.1e-2 .. 10, color = red, view = [0 .. 10, -30 .. 30, -40 .. 40]):

display(DispIntersecting, DispIntersection, axes = boxed, scaling = constrained);

Can somebody help me to find the solution?

I think there is something wrong with the definition of bvw1. If I use dsolve (in soln) with only bvw as Initial Condition,

I get a solution but if I also insert bvw1 as an Initial condition soln won't appear.

Here's what's written in the image:

'Imagine the course of a planet around a star with L=0.5 and e=0.7'

Solve Keppler's differential equation with Initial Conditions:'

Of course, in Maple. Source http://www.mmf.lnu.edu.ua/

Dear all,

I am making a mpl file which looks like below:

font_size := thickness = 4, font = [bold, "TimesNewRoman", 30], labelfont = [bold, "TimesNewRoman", 30], axis[1] = [thickness = 3], axis[2] = [thickness = 3], captionfont = [bold, "TimesNewRoman", 30], legendstyle = [font = [bold, "TimesNewRoman", 30]];

I  export it as mpl, but when I want to read it in a worksheet, I get the error below:

Error, on line 1, syntax error, character `?` unexpected:
<?xml version="1.0" encoding="UTF-8"?>
^
Error, while reading ``/Users/..../font_size.mpl``

This is just a simple mpl, I get this error for all mpl s that I want to read!

Can somebody help me with this please? It is really annoying :(

abc
bca

if starting position is at first position,

i have to write a very long substitution, and need a temporary variable t and t2

subs(t2=c,subs(t=a,subs(a=b,subs(b=t2,subs(c=t,a*b*c+a*b)))));

if starting position is at second position, it will be

abc
acb

is there any simple method to do this circular shift of variables for a polynomial

I am able to get unlimeted numbers of equations describing my system. These equations are generally relate quotients of multivariate polynomials. Each additional equation I get is generally less than twice the length of the last, and it is not always the case that an equation is independant of the previous equations. Although I can get unlimited numbers of equations describing the system, it is not overdetermined.

I am interested in solving these equations for their variables. There are about 30 cases I am working on, the smallest number of evariables is six, the largest would be twenty.

I want to be able to solve these equations in the minimal time possible. But I don't understand the function solve well enough to do so.

How do I choose the equations to minimise the time taken for the command solve to proccess them?
How does the command solve work?

particularly:

1. if I process the command solve([Eq1,Eq2,Eq3...Eqn],variables) would the command solve([Eq[1],Eq[2],Eq[3]...Eq[n],Eq[n+1]],variables) take longer if Eq[n+1] is not indipendant of the previous equations? 
2. Is there a way of checking whether Eq[n+1] is independant of the previous vequations, fast enough for it to be useful to check the equations before they are processed?
3. Does the ordering of the equations affect the speed of solve?
4. Is there a way of pre processing the equations before they are put into solve that will save it time? (for example factorising them, simplifying them etc...)

Hello,

I'm trying to solve the integral u/(1-u) with Maple and noticed that it returned a result that doesn't accord to the solution I found by hand or the solution from WolframAlpha. This is a screenshot of the weird behaviour:

Does Maple do any weird conversions? Or did I do something wrong or is Maple wrong?

Thanks in advance,

Hello everyone!

Suppose, we have a differential polynomial

P := u(x) + (D@@2)(u)(x)

Given this, I am looking for a procedure which gives coefficients depending on the order given as input, for example, lets say, procedure name is fun_coeff which depends on two parameters, original polynomial P and the order n, then

fun_coeff(P, 3) should give [1, 0, 1, 0]

where each entry corresponds to the coeff of

 [u(x), (D@@1)(u)(x), (D@@2)(u)(x), (D@@3)(u)(x)]

in the polynomial, similarly

fun_coeff(P,4) should give [1, 0, 1, 0, 0]

corresponding to

 [u(x), (D@@1)(u)(x), (D@@2)(u)(x), (D@@3)(u)(x), (D@@4)(u)(x) ]

Thank you all for your time :)

Dear all

I have downloaded a third party Maple package from following link:

http://cpc.cs.qub.ac.uk/summaries/AEQP_v1_0.html

The distribution format is  "tar.gz", but I don't know what to with this format, that how should I load this package into Maple library. I have window 8.1.

If anybody have an idea about this format "tar.gz" please help me out.

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Hi

i am working on a simple control project.i should make a transfer function from the below equation (as an example):

 (s^2*m1+I1*(s^2)*(1/r^2))*x(s)+(-(1/2)*s^2*b*m1-s^2*m1*r-I1*(s^2)*(1/r)+m1*g)*theta(s)+(1/2)*theta(0)*s*b*m1+theta(0)*s*m1*r-x(0)*s*m1+(1/2)*(D(theta))(0)*b*m1+(D(theta))(0)*m1*r-(D(x))(0)*m1+I1*s*theta(0)*(1/r)+I1*(D(theta))(0)*(1/r)-I1*s*x(0)*(1/r^2)-I1*(D(x))(0)*(1/r^2) = 0

in order to make the T.F i have to make " x(s)/theta(s) " on the lhs and put the rest on the rhs. i have no idea how to do so.(the underlined part involves no x(s) and theta(s). obviously a theta(s) will appear on the rhs in the one of the denominators which doesn't matter.)

please help me with this problem.since i don't know what the commands are called in english, i failed in searching for it.

thanks.

hi.please remove error in attached file

thanks...gfhf.mw