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Hello every one.

I want to do some tensor computations in maple in a specified coordinate system but I don't know how! As an example I ask the follwing question.

Consider a second order symmetric tensor "A". I want to compute the components of "curl(curl(A))" in cylinderical coordinates. How should I do this in maple?

This is related to a famous equation in elasticity known as "small strain compatibility" equation.

Thanks for the help

In Maple15, in spherical coordinates, what statements plot covariant vector [1,r^2*sin(theta),sin^2(theta)] and its contravariant equivalent [1,sin(theta),1/r^2] within their respective bases vectors. Assume r = 2, theta = pi/6 and phi = pi/4. I presume each plot will display the same space vector.

The Maple help definition for spherical coordinates uses the triple (r, φ, θ) (Note the ordering!!) with φ in the range 0..Π and θ in the range 0..2Π. This means that the second entry in the triple is the zenith angle (latitude) and the third entry in th triple is the azimuth angle (longitude). This is confirmed by the relation to cartesian coordinates stated on the definition page as
x= r sin(φ) cos(θ)

y= r sin(φ) sin(θ)

z= r cos(φ)

However the help page for coords has spherical polars defined by the triple (u, v, w), with the relation to cartesian coordinates given as

x= u cos(v) sin(w)

y= u sin(v) sin(w)

z= u cos(w)

which suggests that this time it is the third entry in the triple (ie w) which is the zenith angle (latitude), with the second entry being the azimuth (longitude).

My simple-minded attempt to check which of these interpretations is correct is shown in the attached worksheet. This seems to confirm that the MapleHelp definitions page is correct and the help/coords page is incorrect - or am I missing something??

Download posVecChk.mw

 

 

Hi all,

 

I'm trying to initialize a 4-vector and promote it as a tensor in Maple 16 using the Physics package. My attempt so far has been unsuccessful :) 

This 4-vector needs to be a function of the coordinates X. What I'm trying to do is to allocate terms by terms my tensor as indicated in my attempt below. Everything looks fine (the query is telling me F2 is a tensor) but when I'm trying to compute contravariant quantities, I can't obtain an answer for a given coordinate (here F2[~n](X) is not equal du v_1(X))... Finally, the last line, computing the sum over all the indexes is not giving me an answer... 

 

restart; with(Physics):
Setup(mathematicalnotation=true):
Setup(coordinatesystems = cartesian):
Define();
g_[]:

F2[1]:=(x,y,z,t)->v_1(X);F2[2]:=(x,y,z,t)->v_2(X);F2[3]:=(x,y,z,t)->v_3(X);F2[4]:=(x,y,z,t)->1;

Define(F2[mu](X));

Define(F2,query);

F2[1](X);F2[2](X);

F2[~1](X);F2[~2](X);

F2[~nu].F2[nu]

 

Surely, something is wrong in my way of defining a spacetime tensor. What would be the solution to obtain a decent contravariant F2?

 

Another more or less related question would be to know how to compute a taylor expansion of a function itself i.e. 1/(1+f(X))~ 1-f(X) since f<<1? Would there be a built-in fonction to do such a thing?

 

Many thanks!

 

Vincent

Hello,

I have been trying to compute the analytical solution of two dimensional diffusion equation with zero neumann boundary conditions (no-flux) in polar coordinates using the solution in Andrei Polyanin's book. When I use 2d Gaussian function as initial condition, i cannot get the result. If I use some nicer function like f(r,phi)=1-r; there is no problem.  

Any idea why this happens? or any suggestion to compute the analytical solution?

Thanks!

HB 

M := Matrix([[3.83170597020751, 7.01558666981561, 10.1734681350627, 13.3236919363142, 16.4706300508776], [1.84118378134065, 5.33144277352503, 8.53631636634628, 11.7060049025920, 14.8635886339090], [3.05423692822714, 6.70613319415845, 9.96946782308759, 13.1703708560161, 16.3475223183217], [4.20118894121052, 8.01523659837595, 11.3459243107430, 14.5858482861670, 17.7887478660664], [5.31755312608399, 9.28239628524161, 12.6819084426388, 15.9641070377315, 19.1960288000489], [6.41561637570024, 10.5198608737723, 13.9871886301403, 17.3128424878846, 20.5755145213868], [7.50126614468414, 11.7349359530427, 15.2681814610978, 18.6374430096662, 21.9317150178022], [8.57783648971407, 12.9323862370895, 16.5293658843669, 19.9418533665273, 23.2680529264575], [9.64742165199721, 14.1155189078946, 17.7740123669152, 21.2290626228531, 24.5871974863176], [10.7114339706999, 15.2867376673329, 19.0045935379460, 22.5013987267772, 25.8912772768391], [11.7708766749555, 16.4478527484865, 20.2230314126817, 23.7607158603274, 27.1820215271905]]):

c := 10:

A := 5:

w := proc (r, phi, t) options operator, arrow; int(int(f(xi, eta)*G(r, phi, xi, eta, t)*xi, xi = 0 .. 5), eta = 0 .. 2*Pi) end proc:

with(plots):

Warning,  computation interrupted

 

``



Download 2d_soln.mw

Hi,

I need to build a multibody model in MapleSim 6.4 in which with few global parameters I can describe all the other parameters. In other words the final user will enter this few parameters, that are coordinates of specific points, and then the model will calculate all the relative distances on the base of those coordinates.

The problem is that if I apply trigonometric function and square root (like in the screenshot) the model is not calculating any value. Is it possible to make those calculculations?

 

this is the model (don't worry about the nonsense plots, it's because it's not ultimated):

DWS.msim

 

 

Thanks.

 

Let Poly2 denote the vector space of polynomials

(with real coefficients) of degree less than 3.

Poly2 = {a1t^2+ a2 t+ a3 |a1; a2; a3 €R}

You may assume that {1,t; t^2}is a basis for Poly2.

(1) Show that L1 = {t^2 + 1; t-2 ; t + 3}and L2 = {2 t^2 + t; t^2 + 3; t}

are bases for Poly2.

(2) Let v = 8t^2- 4t + 6 and w = 7t^2- t + 9. Find the coordinates for

v and w with respect to the basis L1 and with respect to the basis L2

Hello,

I have a question concerning the initial conditions.

One interest of MapleSim is to enable to model a multibody systems with different kinds of coordinates : namely relative and absolute coordinates.

For a complex system, it seems to me that the determination of the initial position is not an easy task.

Consequently, I wonder if it is possible to determine the initial positions for the system in absolute coordinates thanks to the knowledge of the initials positions in relative coordinates.

In other words, I have already determined the initial positions of my system in relative coordinates but as I would like to simulate my system with absolute coordinates. I wonder if I can have a process to deduct the initial positions for the absolute coordinates for the initial positions in relative coordinates.

Thanks a lot for your ideas and help.

In the book "Challenges in Geometry" of the author Christopher J. Bradley at p. 32, the triangle with three sides a := 136, b := 170, c := 174 has three medians ma := 158, mb := 131, mc := 127. I checked

restart:

a:=2*68;

b:=2*85;

c:=2*87;

ma:=sqrt((b^2+c^2)/2-a^2/4);

mb:=sqrt((a^2+c^2)/2-b^2/4);

mc:=sqrt((b^2+a^2)/2-c^2/4);

Now I want to find coordinates of vertices of a triangle like that (in plane). I tried

restart;
DirectSearch:-SolveEquations([(x2-x1)^2+(y2-y1)^2 = 136^2,
(x3-x2)^2+(y3-y2)^2 = 170^2, (x3-x1)^2+(y3-y1)^2 = 174^2], {abs(x1) <= 30, abs(x2) <= 30, abs(y1) <= 30, abs(y2) <= 30, abs(x3) <= 30, abs(y3) <= 30}, assume = integer, AllSolutions, solutions = 5);

but my computer runs too long. I think, there is not a triangle with integer coordiantes. 

How can I get  a triangle  with coordinates of vertices are rational numbers?

 

Level: Idiot (Me)

I have a matrix of 3 columns and lots of rows M

  • First column is latitude in degrees
  • Second column is longitude in degrees
  • Third column is data

So I set lambda:=M(..,1) and phi_g:=M(..,2) giving me two column vectors.

I want to convert lambda and phi_g to polar coordinates theta and phi

theta:=90-lambda produces "Error, (in rtable/Sum) invalid arguments"

WHY?

I also want to convert phi_g to phi where phi=phi_g when phi_g is 0...180 and phi=phi_g +360 when phi_g <0

How do I create a conditional function like this?

Hallo,

Is there a fairly straightforward method for obtaining an array of coordinates from an implicit equation? I have an ellipse defined implicitly (by a horrendously involved expression) and can't figure out how to extract a set of coordinates from an implicitplot. I'm reluctant to use seq and fsolve with a fixed stepsize.

Would be grateful for some insight!

Thanks.

Weirp

I am making something for my math prof as a token of appreciation. It is a spiral slide rule designed to fit in a watch face, but it can be made bigger. I threw together a program to show what I mean:

https://www.khanacademy.org/cs/spiral-slide-rule/5831476045152256

I was told that maple can be used to create things like this, but since I am new to the software, I really don't know how to put something like this together. The best I could do was to make a spiral in polar coordinates, and plot my lists of points on top of it. However, this does not annotate the points and I can't make nice looking lines (the only point options in polarplot are stars, crosses, and other things that aren't what Im looking for).

Does anyone know how I could model this, and then export it as a format where I might be able to send to a printer to get it printed on specialty paper?

Thank you very much

 

EDIT: I uploaded my plot here, in case you want to see it:

http://randomwrites.wikispaces.com/file/view/mediumspiral1.bmp/508315780/mediumspiral1.bmp

In that attached file is a multip step problem that involves graphing a right circular cylinder using transtion matrices and orthonormal basis. I have completed the hole question minus the very last part which is asking for new parametric equations for the cylinder if its center point is located at (-2, 10, 3) instead of the origin.

Any ideas on how to do this will be greatly appreciated.

 

question_4.mw

Please advise as to the proper coding entries needed in the triple integration palette  to transform from the Cartesian placeholders x;y;z to spherical coordinates rho; theta; phi so that the triple integration palette can be used in spherical coordinmates. Dr. Lopez alreadyb has a standalone template which does this but I would like to set a palette option for spherical  calculations.

 

Thanx

I have a diff equation in cartesian coordinates I need to transform to a certain cylindrical system. The de looks like this:

I define my new system with addcoords like this:

addcoords('AccCylinder',[r,theta,y],[r*cos(theta),y,r*sin(theta)]); # Note: y is the longitudinal axis here!

and also do

VectorCalculus:-AddCoordinates('AccCylinder'[r,theta,y],[r*cos(theta),y,r*sin(theta)],overwrite) assuming r >= 0;

and note that I had to overwrite as my system was already known, so maybe addcoords is reduncant(?)

I then do the transformation ("(7)" is the label of my above de):

changecoords((7),[x,y,z],AccCylinder,[r,theta,y]);

and get

This may be correct, but it has the expressions in the differentials, which diff does not know how to handle. I need to convert things like diff(xpr,r*cos(theta)) = diff(xpr,x)*diff(x,r) where x would be r*cos(theta). I can do this "by hand", but that seems overly tedious and error-prone. Somehow I'd expect the coordinate transforms to be able to do this but I can't figure out how.

Any idea?

Thanks,

Mac Dude,

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