restart;  
with(geometry):  
with(plots):  
_EnvHorizontalName = 'x':  _EnvVerticalName = 'y':
point(A, -1, 9):                                                                                                       
point(B, -5, 0):
point(C, 6, 0):
triangle(ABC,[A,B,C]):
midpoint(M1,A,C): midpoint(M2,B,C):midpoint(M3,A,B):
rotation(J, C, Pi/2, 'counterclockwise', M1):triangle(AJC,[A,J,C]):
rotation(Ii, C, Pi/2, 'counterclockwise', M2):triangle(BIC,[B,Ii,C]):
rotation(K, A, Pi/2, 'counterclockwise', M3):triangle(AKB,[A,K,B]):
midpoint(O1,K,J): coordinates(O1):
midpoint(O2,A,Ii): coordinates(O2):  
poly:=[coordinates(A),coordinates(J),coordinates(Ii),coordinates(K)]:   

display(draw([A(color = black, symbol = solidcircle, symbolsize = 12), 
B(color = black, symbol = solidcircle, symbolsize = 12), 
C(color = black, symbol = solidcircle, symbolsize = 12), 
J(color = black, symbol = solidcircle, symbolsize = 12), 
polygonplot(poly,color = "DarkGreen", transparency = 0.5),
ABC(color = red ),
BIC(color = green),
AKB(color = grey),
AJC(color =blue)]),
textplot([[coordinates(A)[], "A"],[coordinates(J)[], "J"],[coordinates(Ii)[], "I"],   
[coordinates(B)[], "B"], [coordinates(K)[], "K"], 
[coordinates(C)[], "C"]], 
align = [above, right]),  axes = none);
Error, (in geometry:-draw) the option must be of type equation or name. I don't see how to correct this error/
 

Hi, I have the equation below and I'm trying to use "collect " to get the coefficient of dudy^2 and dvdx^2. I tried

aa1 := subs({gamma[1] = alpha[3]-alpha[2], gamma[2] = alpha[6]-alpha[5]}, collect(aa, {dudy^2, dvdx^2}))

But is not collecting the coefficient of dvdx^2. Please can someone help?

aa := alpha[1]*(cos(theta(x, y, t))^4*dudx^2+2*cos(theta(x, y, t))^3*dudx*sin(theta(x, y, t))*dvdx+2*cos(theta(x, y, t))^3*dudx*sin(theta(x, y, t))*dudy+2*cos(theta(x, y, t))^2*dudx*sin(theta(x, y, t))^2*dvdy+cos(theta(x, y, t))^2*sin(theta(x, y, t))^2*dvdx^2+2*cos(theta(x, y, t))^2*sin(theta(x, y, t))^2*dvdx*dudy+2*cos(theta(x, y, t))*sin(theta(x, y, t))^3*dvdx*dvdy+cos(theta(x, y, t))^2*sin(theta(x, y, t))^2*dudy^2+2*cos(theta(x, y, t))*sin(theta(x, y, t))^3*dudy*dvdy+sin(theta(x, y, t))^4*dvdy^2)+(alpha[2]+alpha[3]+gamma[2])*(-dudx*cos(theta(x, y, t))*sin(theta(x, y, t))*thetadot-(1/2)*dudx*cos(theta(x, y, t))*sin(theta(x, y, t))*dudy+(1/2)*dudx*cos(theta(x, y, t))*sin(theta(x, y, t))*dvdx+(1/2)*cos(theta(x, y, t))^2*dvdx*thetadot-(1/4)*cos(theta(x, y, t))^2*dvdx^2+(1/2)*cos(theta(x, y, t))^2*dudy*thetadot+(1/4)*cos(theta(x, y, t))^2*dudy^2-(1/2)*sin(theta(x, y, t))^2*dvdx*thetadot+(1/4)*sin(theta(x, y, t))^2*dvdx^2-(1/2)*sin(theta(x, y, t))^2*dudy*thetadot-(1/4)*sin(theta(x, y, t))^2*dudy^2+dvdy*sin(theta(x, y, t))*cos(theta(x, y, t))*thetadot-(1/2)*dvdy*sin(theta(x, y, t))*cos(theta(x, y, t))*dvdx+(1/2)*dvdy*sin(theta(x, y, t))*cos(theta(x, y, t))*dudy)+alpha[4]*(dudx^2+(1/2)*dvdx^2+dvdx*dudy+(1/2)*dudy^2+dvdy^2)+(alpha[5]+alpha[6])*(cos(theta(x, y, t))^2*dudx^2+2*cos(theta(x, y, t))*dudx*((1/2)*dvdx+(1/2)*dudy)*sin(theta(x, y, t))+cos(theta(x, y, t))^2*((1/2)*dvdx+(1/2)*dudy)^2+2*cos(theta(x, y, t))*((1/2)*dvdx+(1/2)*dudy)*dvdy*sin(theta(x, y, t))+sin(theta(x, y, t))^2*((1/2)*dvdx+(1/2)*dudy)^2+sin(theta(x, y, t))^2*dvdy^2)+gamma[1]*(sin(theta(x, y, t))^2*thetadot^2+sin(theta(x, y, t))^2*dudy*thetadot-sin(theta(x, y, t))^2*dvdx*thetadot+(1/4)*sin(theta(x, y, t))^2*dudy^2-(1/2)*sin(theta(x, y, t))^2*dudy*dvdx+(1/4)*sin(theta(x, y, t))^2*dvdx^2+cos(theta(x, y, t))^2*thetadot^2-cos(theta(x, y, t))^2*dvdx*thetadot+cos(theta(x, y, t))^2*dudy*thetadot+(1/4)*cos(theta(x, y, t))^2*dvdx^2-(1/2)*cos(theta(x, y, t))^2*dudy*dvdx+(1/4)*cos(theta(x, y, t))^2*dudy^2)+xi*(cos(theta(x, y, t))^2*dudx+2*cos(theta(x, y, t))*sin(theta(x, y, t))*((1/2)*dvdx+(1/2)*dudy)+sin(theta(x, y, t))^2*dvdy);

hi every body 

please How to write these solutions in fraction form

Download problem1.mw

How to get the series and plot.I got this error.

TL.mw

At the end of 2021, Mathematica added capability for integration:

https://blog.wolfram.com/2021/12/13/new-in-13-symbolic-numeric-computation/

Trying this in Maple, it does not look like it can solve it:

My questions are then, is there a way to get Maple to solve this? Does Maplesoft actively work on solving more integral types and is it likely support for this integral will be added at some point?

In Hansen's book "A table of Series and Products" there is an unsourced listed identity 43.6.1:

S1 := Sum(2^k*tanh(2^k*x), k = 0 .. n - 1) = 2^n*coth(2^n*x) -   coth(x)

It appears to be to be numerically correct, and can be obtained from a second identity

S6:=Sum(2^(-k)*tanh(2^(-k)*x), k = 1 .. n) =- 2^(-n)*coth(2^(-n)*x) + coth(x)

by "summing in reverse order". S6 is also numerically correct.

Starting from S1 and reversing the order, gives

S2 := Sum(2^(n - k - 1)*tanh(2^(n - k - 1)*x), k = 0 .. n - 1) = 2^n*coth(2^n*x) - coth(x)

Then shifting the index gives

 S3 := Sum(2^(-k)*tanh(2^(-k)*x), k = -n + 1 .. 0) =  2^n*coth(2^n*x) - coth(x)

both of which test OK numerically since n>0. Now it begins to get strange.

In S3, let n->-n (highly illegal) to obtain

S4 := Sum(2^(-k)*tanh(2^(-k)*x), k = n + 1 .. 0) = 2^(-n)*coth(2^(-n)*x) - coth(x)

and, surprisingly Maple tests S4 to be correct numerically.

Here's how that happened. The lower limit of the S4 sum exceeds the upper limit, so there are three possible ways to interpret this:

1. In standard mathematical usage, this is usually set to zero by caveat;

2. In a practical sense, one might expect the order of summation to be irrelevant so S4 is the same as

S5:= Sum(2^(-k)*tanh(2^(-k)*x), k = 0 .. n + 1) = 2^(-n)*coth(2^(-n)*x) - coth(x)

but that fails numerically.

3. Maple has a built-in algorithm (see the help page on "Sum") that interprets

Sum(f(k), k = m .. n) = -Sum(f(k), k = n + 1 .. m - 1)

when m exceeds n and that is why S4 works numerically and, when applied to S4 gives the result S6.

So where did Maple's prescription come from, and what is it's justification?

Thank you

This is from a graph G with vertex set {0,1,2,3,4,5,6,7,8,9} always labelled from {0,1,2,3,...,n-1}

Now to Split the List L to sublists like this

Now going through L list in details (All my lists will be like this only}

Sublist L1 will have all those from first in this firstly we see  edge {0,1} we take all those which take {0,1} in sequencial manner and put in sublist L1

Now as we proceed we see the next starting edge is {0,3} so their is none with {0,2} so we pick all those with {0,3} in the first sequencial manner and put in sublist L2

Similiar we pick all sublist from this with first element unique in the sequential manner and make a list of lists

Lk:=[L1,L2,L3,L4,....]

Code until L

which was done

Toy_code_(1).mw

Now again proceed we can observe we can see their none with {0,4} next is only with {0,5} 

Then i need write a function F which takes a List say L1 returns

all possible 2 element permutions from L1 list say [S1,S2] and [S2,S1] like that all possible as order also matter where that set is positioned

Given Graph G and Graph H how to store the list of all the various Subgraphs of Shape H in G with their vertex labels.

I have write a code but it may not be the optimal way to do it and it may be time consuming for even medium graphs too

Any way to optimize

Let From this list of subgraphs H of say [G0,G1,G2,.....] we pick NumberOfVertices(G) Graphs
say H= [H0,H1,H2,H3,....,Hn]

i) Now any two graph Hi,Hj have excatly one edge interection if (i,j) is an edge in G

 ii) An edge in  the list H will occur exactly twice only in the entire list of graphs

If these two conditions satifies all such H I need to save

Toycode until picking graphs

From that how to pick lists which satify conditions i) and ii) is another need help

Toy_code.mw

Please excuse my thickness if any.

In the following IF statement:

if 1 <> 2 then
    OuterThen;
    if 1 <> 2 then
        InnerThen;
    else
        InnerElse;
    end if;    
else
    OuterElse;
end if;

I expect the output:

OuterThen
InnerThen

but I only get:

OuterThen

Why?

In practice, I need calculate the (principal) squareroot of some suitable large matrix exactly (so the desired result should not involve floating-point numbers, otherwise the decomposition will be of no theoretical value as a certificate …). But I find it difficult to do so in Maple. (What about the Efficient Computations - Maple Help (maplesoft.com)?)
Below are some matrices: (They are not contrived academic examples.) 

M__4 := <1,0,-2,0,1,0,0,0,0,0,0,0,0,0,0,-2,0,2,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0|0,3,0,-6,0,3,0,0,0,0,0,0,0,0,0,0,-3,0,3,0,-3,0,0,0,0,0,0,0,0,6,0,0,0,0,-3,0|-2,0,7,0,-8,0,3,0,0,0,0,0,0,0,0,1,0,-7,0,2,0,0,0,0,0,0,4,0,3,0,0,0,0,-3,0,0|0,-6,0,13,0,-8,0,1,0,0,0,0,0,0,0,0,6,0,-4,0,4,0,0,0,0,0,0,-1,0,-11,0,0,0,0,6,0|1,0,-8,0,13,0,-6,0,0,0,0,0,0,0,0,4,0,8,0,-1,0,0,0,0,0,0,-11,0,-6,0,0,0,0,6,0,0|0,3,0,-8,0,7,0,-2,0,0,0,0,0,0,0,0,-3,0,-1,0,1,0,0,0,0,0,0,2,0,4,0,0,0,0,-3,0|0,0,3,0,-6,0,3,0,0,0,0,0,0,0,0,-3,0,-3,0,0,0,0,0,0,0,0,6,0,3,0,0,0,0,-3,0,0|0,0,0,1,0,-2,0,1,0,0,0,0,0,0,0,0,0,0,2,0,-2,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0|0,0,0,0,0,0,0,0,3,0,0,0,3,0,-3,0,0,0,0,0,0,-6,0,-3,0,6,0,0,0,0,3,0,-3,0,0,0|0,0,0,0,0,0,0,0,0,4,0,0,0,-2,0,0,0,0,0,0,0,0,-6,0,6,0,0,0,0,0,0,-2,0,0,0,0|0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,-2,0,-1,0,0,0,0,2,0,2,0,0,-1|0,0,0,0,0,0,0,0,0,0,0,12,0,-6,0,0,0,0,0,0,0,0,-6,0,-6,0,0,0,0,0,0,6,0,0,0,0|0,0,0,0,0,0,0,0,3,0,0,0,3,0,-3,0,0,0,0,0,0,-6,0,-3,0,6,0,0,0,0,3,0,-3,0,0,0|0,0,0,0,0,0,0,0,0,-2,0,-6,0,4,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,-2,0,0,0,0|0,0,0,0,0,0,0,0,-3,0,0,0,-3,0,3,0,0,0,0,0,0,6,0,3,0,-6,0,0,0,0,-3,0,3,0,0,0|-2,0,1,0,4,0,-3,0,0,0,0,0,0,0,0,7,0,-1,0,2,0,0,0,0,0,0,-8,0,-3,0,0,0,0,3,0,0|0,-3,0,6,0,-3,0,0,0,0,0,0,0,0,0,0,3,0,-3,0,3,0,0,0,0,0,0,0,0,-6,0,0,0,0,3,0|2,0,-7,0,8,0,-3,0,0,0,0,0,0,0,0,-1,0,7,0,-2,0,0,0,0,0,0,-4,0,-3,0,0,0,0,3,0,0|0,3,0,-4,0,-1,0,2,0,0,0,0,0,0,0,0,-3,0,7,0,-7,0,0,0,0,0,0,-2,0,8,0,0,0,0,-3,0|-1,0,2,0,-1,0,0,0,0,0,0,0,0,0,0,2,0,-2,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0|0,-3,0,4,0,1,0,-2,0,0,0,0,0,0,0,0,3,0,-7,0,7,0,0,0,0,0,0,2,0,-8,0,0,0,0,3,0|0,0,0,0,0,0,0,0,-6,0,-1,0,-6,0,6,0,0,0,0,0,0,13,0,8,0,-11,0,0,0,0,-8,0,4,0,0,1|0,0,0,0,0,0,0,0,0,-6,0,-6,0,6,0,0,0,0,0,0,0,0,12,0,-6,0,0,0,0,0,0,0,0,0,0,0|0,0,0,0,0,0,0,0,-3,0,-2,0,-3,0,3,0,0,0,0,0,0,8,0,7,0,-4,0,0,0,0,-7,0,-1,0,0,2|0,0,0,0,0,0,0,0,0,6,0,-6,0,0,0,0,0,0,0,0,0,0,-6,0,12,0,0,0,0,0,0,-6,0,0,0,0|0,0,0,0,0,0,0,0,6,0,-1,0,6,0,-6,0,0,0,0,0,0,-11,0,-4,0,13,0,0,0,0,4,0,-8,0,0,1|1,0,4,0,-11,0,6,0,0,0,0,0,0,0,0,-8,0,-4,0,-1,0,0,0,0,0,0,13,0,6,0,0,0,0,-6,0,0|0,0,0,-1,0,2,0,-1,0,0,0,0,0,0,0,0,0,0,-2,0,2,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0|0,0,3,0,-6,0,3,0,0,0,0,0,0,0,0,-3,0,-3,0,0,0,0,0,0,0,0,6,0,3,0,0,0,0,-3,0,0|0,6,0,-11,0,4,0,1,0,0,0,0,0,0,0,0,-6,0,8,0,-8,0,0,0,0,0,0,-1,0,13,0,0,0,0,-6,0|0,0,0,0,0,0,0,0,3,0,2,0,3,0,-3,0,0,0,0,0,0,-8,0,-7,0,4,0,0,0,0,7,0,1,0,0,-2|0,0,0,0,0,0,0,0,0,-2,0,6,0,-2,0,0,0,0,0,0,0,0,0,0,-6,0,0,0,0,0,0,4,0,0,0,0|0,0,0,0,0,0,0,0,-3,0,2,0,-3,0,3,0,0,0,0,0,0,4,0,-1,0,-8,0,0,0,0,1,0,7,0,0,-2|0,0,-3,0,6,0,-3,0,0,0,0,0,0,0,0,3,0,3,0,0,0,0,0,0,0,0,-6,0,-3,0,0,0,0,3,0,0|0,-3,0,6,0,-3,0,0,0,0,0,0,0,0,0,0,3,0,-3,0,3,0,0,0,0,0,0,0,0,-6,0,0,0,0,3,0|0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,2,0,1,0,0,0,0,-2,0,-2,0,0,1>:
M__3 := 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Unexpectedly, LinearAlgebra[MatrixPower] does not work well. For instance: 
 

Download performance_of_`sqrtm`.mw

The last six numerical experiments all fail to compute the square root. (Accordingly, here it is impossible to convert each numeric element to one of the "simplest" algebraic numbers that approximates it well.) As you can see, if I execute them directly (i.e., without converting each entry to the nearest floating-point value), the elapsed time required to run the procedure is still unacceptable! (Note that they can be evaluated symbolically in fact.) Is this a bug of Maple? And how do I get the desired results in Maple efficiently?

Hi,

I want to illustrate the intersection between The ellipsoid   "x^2/4 + y^2/9 + z^2/16 = 1"and the plane  "z = 3"

Normally, I should obtain an ellipse G1, but its graph does not match the graph obtained ? Thanks for your ideas
 

G2Exercice1.mw

How to solve this RLC Electric Circuit  with this initial conditions,I couldn't plot this equation.Help me to do this problem.

EC-2.mw

Let y＞1. It can be proved (maybe by hand) that the following four expressions are mathematically equivalent: 

assume(y > 1); # Assumption!
expr := [0, 0, 0, 0]: # Preallocation.
expr[1] := exp(y*LambertW(ln(y))):
expr[2] := (ln(y)/LambertW(ln(y)))^y:
expr[3] := eval(x^(x^x), x = exp(LambertW(ln(y)))):
expr[4] := eval(x^(x^x), x = ln(y)/LambertW(ln(y))):

But unfortunately, when I tried to simplify expr_i - expr_j (symbolically), I just got: 

seq(seq(ifelse(j <> i, [i, j, verify(expr[j], expr[i], equal)], NULL), j = 1 .. numelems(expr)), i = 1 .. numelems(expr)); # is(expr[j] = expr[i]) does not work as well.
 = 
   [1, 2, FAIL], [1, 3, FAIL], [1, 4, FAIL], [2, 1, FAIL], 

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

     [3, 4, true], [4, 1, FAIL], [4, 2, FAIL], [4, 3, true]

In other words, Maple can only determine that expr[3] = expr[4].
One may check that, for example, 

MmaTranslator:-Mma:-Chop([seq](seq(evalhf(subs(y = log10(rand()), expr[i] - expr[j])), j = 1 .. numelems(expr)), i = 1 .. numelems(expr)), 2^(-26));
 = 
        [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

However, the desired approach is simplifying them symbolically. Is there a way to do so in Maple?

Hi, I'm trying to add the display of graphs G1, G2, and G3 (representing the intersections -AnimCompl file-) to my three animations (G5Ex2-file). I've tried using the background option and the display command, but without success. Thank you for your help.

AnimCompl.mw

G5Ex2.mw

I want to put one option of tetrahedron SABC with AB=c, BC=a, AC=b, SA=m, SB=n, SC=p. I tried
restart;
with(Student:-MultivariateCalculus);
A := [0, 0, 0];
B := [c, 0, 0];
C := [(-a^2 + b^2 + c^2)/(2*c), sqrt((a + b - c)*(a - b + c)*(-a + b + c)*(a + b + c))/(2*c), 0];
S := [x, y, z];
solve([distance(S, A) = m, distance(S, B) = n, distance(S, C) = p], [x, y, z]);

I got warning, solutions may have been lost. How can I solve that system of equations?