Maple Questions and Posts

These are Posts and Questions associated with the product, Maple

Hi,
Can anyone tell me what's going wrong with int(Heaviside(f-1/2), x=0..1, y=0..1); ?
 

restart:

f := x*y:

JH := int(Heaviside(f-1/2), x=0..1, y=0..1);  #???
                               1
                               -
                               4

 JP := int(piecewise(x*y>1/2, 1, 0), x=0..1, y=0..1); 
                          1   1      
                          - - - ln(2)
                          2   2      

# integrate "1" over the domain where f > 1/2

S := solve(f=1/2, x):
JS := int(1, x=S..1, y=1/2..1);
                          1   1      
                          - - - ln(2)
                          2   2      

Thanks in advance
 

При использовании «объединения» нумерация не сохраняется. Непонятно, по каким алгоритмам «объединение» применяется к тому или иному месту. Почему sistema [1] не равно GRAN [1]? Как я могу это исправить?

PS Прошу прощения за грамматические ошибки, английский плохо знаю

1. mws
 

рестарт;

 

 

B:=matrix([[1,        0,        1,        0,        0,        0,        0,        0,        0,        0,        0,        0,        0,        0,        0,        0],
[-1,        -1,        0,        0,        0,        0,        0,        0,        0,        0,        0,        0,        0,        0,        0,        0],
[0,        0,        -1,        0,        -1,        0,        0,        0,        0,        0,        0,        0,        0,        0,        0,        0],
[0,        1,        0,        1,        0,        0,        0,        0,        0,        0,        0,        0,        0,        0,        0,        0],
[0,        0,        0,        0,        1,        1,        0,        1,        0,        0,        0,        0,        0,        0,        0,        0],
[0,        0,        0,        -1,        0,        -1,        -1,        0,        0,        0,        0,        0,        0,        0,        0,        0],
[0,        0,        0,        0,        0,        0,        1,        0,        1,        0,        0,        0,        0,        0,        0,        0],
[0,        0,        0,        0,        0,        0,        0,        -1,        0,        -1,        0,        0,        0,        0,        0,        0],
[0,        0,        0,        0,        0,        0,        0,        0,        -1,        0,        -1,        -1,        0,        0,        0,        0],
[0,        0,        0,        0,        0,        0,        0,        0,        0,        1,        1,        0,        1,        0,        0,        0],
[0,        0,        0,        0,        0,        0,        0,        0,        0,        0,        0,        1,        0,        1,        0,        0],
[0,        0,        0,        0,        0,        0,        0,        0,        0,        0,        0,        0,        -1,        0,        -1,        0],
[0,        0,        0,        0,        0,        0,        0,        0,        0,        0,        0,        0,        0,        -1,        0,        -1],
[0,        0,        0,        0,        0,        0,        0,        0,        0,        0,        0,        0,        0,        0,        1,        1]
]):

E:=16:V:=14;

V: = 14

(1)

G:=2*E; êîëè÷åñòâî ãðàíè÷íûõ óñëîâèé

G: = 32

(2)

l:=[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]:

 

for i to E do
De[i]:=-diff(phi[i](s[i]),s[i]$2)=He*phi[i](s[i]):  Äó âèäà -(diff(phi[i](s[i]), `$`(s[i], 2))) = H[e]*phi[i](s[i])
od:

for i to E do
FSR[i]:=subs(s[i]=x,dsolve(De[i],phi[i](s[i]),output=basis)):  Îáùèå ðåøåíèÿ
od:

for i to E do
OBCHEE_RECHIE[i]:=unapply(C[i+i-1]*FSR[i][2]+C[i+i]*FSR[i][1],x): od:

 

for i to E do
PROIZ:=diff(OBCHEE_RECHIE[i](x),x):
PROIZ_OB_RECH[i]:=unapply(PROIZ,x):od:

 

×òåíèå ìàòðèöû èíöèäåíòíîñòè

i0:=1:i1:=1:z:=1:y:=1:

for i from 1 to V do
 y:=1: z:=1:
  for j from 1 to E do
   if B[i,j]=1 then
    x[z]:=j: z:=z+1:
   fi:

   if B[i,j]=-1 then
    x[y]:=j:
    y:=y+1:
   fi:

  od;

  if z>=2 then
   v0[i0]:=[seq(x[s],s=1..z-1)]:
   i0:=i0+1:
  fi:

  if y>=2 then
   v1[i1]:=[seq(x[s],s=1..y-1)]:
   i1:=i1+1:
  fi:
od:

v0:=convert(v0,list):

v1:=convert(v1,list):

 

Ïîäñ÷åò êîëè÷åñòâà ãðàíè÷íûõ óñëîâèé òèïà f(0)

number0:=0:

for i to i0-1 do
if nops(v0[i])=2 then number0:=number0+2; fi:
if nops(v0[i])=3 then number0:=number0+4; fi:
od:

number0:=number0/2; #êîëè÷åñòâî ãðàíè÷íûõ óñëîâèé òèïà f(0)

number0 := 9

(3)

 

Ïîäñ÷åò êîëè÷åñòâà ãðàíè÷íûõ óñëîâèé òèïà f(1)

number1:=0:

for i to i1-1 do
if nops(v1[i])=2 then number1:=number1+2; fi:
if nops(v1[i])=3 then number1:=number1+4; fi:
od:

number1:=number1/2; #êîëè÷åñòâî ãðàíè÷íûõ óñëîâèé òèïà f(1)

number1 := 9

(4)

 

Êîëè÷åñòâî ãðàíè÷íûõ óñëîâèé ñ ïðîèçâîäíûìè

GP:=i0-1+i1-1;

GP := 14

(5)

 

ÍÓËÅÂÛÅ ÓÑËÎÂÈß

i:=1:

while i<number0+1 do
for j to i0-1 do
if nops(v0[j])=2 then
GRAN[i]:=OBCHEE_RECHIE[v0[j,1]](0)-OBCHEE_RECHIE[v0[j,2]](0)=0;
i:=i+1;
fi;
if nops(v0[j])=3 then
GRAN[i]:=OBCHEE_RECHIE[v0[j,1]](0)-OBCHEE_RECHIE[v0[j,2]](0)=0;
GRAN[i+1]:=OBCHEE_RECHIE[v0[j,2]](0)-OBCHEE_RECHIE[v0[j,3]](0)=0;
i:=i+2;
fi;
od; od;

 

ÅÄÈÍÈ×ÍÛÅ ÓÑËÎÂÈß (ÈËÈ îò l)

i:=number0+1:

while i<number0+number1+1 do
for j to i1-1 do
if nops(v1[j])=2 then
GRAN[i]:=OBCHEE_RECHIE[v1[j,1]](l[v1[j,1]])-OBCHEE_RECHIE[v1[j,2]](l[v1[j,2]])=0;
i:=i+1;
fi;
if nops(v1[j])=3 then
GRAN[i]:=OBCHEE_RECHIE[v1[j,1]](l[v1[j,1]])-OBCHEE_RECHIE[v1[j,2]](l[v1[j,2]])=0;
GRAN[i+1]:=OBCHEE_RECHIE[v1[j,2]](l[v1[j,2]])-OBCHEE_RECHIE[v1[j,3]](l[v1[j,3]])=0;
i:=i+2;
fi;
od; od;

 

ÍÓËÅÂÛÅ ÏÐÎÈÇÂÎÄÍÛÅ

i:=number0+number1+1:

while i<number0+number1+nops(v0) do
 for j to nops(v0) do
  for y from 1 to nops(v0[j]) do
   x[y]:=PROIZ_OB_RECH[v0[j,y]](0);
  od;
  m:='m';
  GRAN[i]:=sum(x[m],m=1..nops(v0[j]));
  i:=i+1;
 od;
od;

 

ÍÅÍÓËÅÂÛÅ ÏÐÎÈÇÂÎÄÍÛÅ

i: = число0 + число1 + nops (v0) +1:

в то время как i <G + 1 сделать
 для j до nops (v1) сделать
  для y от 1 до nops (v1 [j]) сделать
   x [y]: = PROIZ_OB_RECH [v1 [j, y]] (l [v1 [j, y]]);
  od;
  m: = 'm';
  ГРАН [i]: = сумма (x [m], m = 1..nops (v1 [j]));
  я: = я + 1;
 od;
od;

 

 

 

новогодний

sistema: = {}:

новенькие

время: = {}:

 

для я от 2 до G делаю

  sistema: = sistema union {GRAN [i]}:

od:

для я от 2 до G делаю

  peremenie: = peremenie union {C [i]}:

od:

 

коэфф: = решить (система, изменение):

коэфф:

 

система [1];

C[2]*He^(1/2)+C[6]*He^(1/2)

(6)

ГРАН [1];

C[1]-C[5] = 0

(7)

 

urav: = subs (koeff, GRAN [1]):

uravnenie: = collect (numer (lhs (normal (urav))), {He, C [1]}):

uravn: = remove (has,%, C [1]):

uravnenie_sobs_chisla: = evalf (algsubs (He = x, uravn)):

ВЕСЕЛЫЕ: = неприменить (uravnenie_sobs_chisla, x):

 

 

 

 

 

 

 

 

 

 

 

 


 

Скачать 1.mws

 

 

Initially I have a system of w(x,y,t) and f(x,y,t). I have a set of two partial differential equations with this system. Later on, the system is modified into w(x,y,t,z) andf(x,y,t,z) where x and z are related due to the presence of boundary layers. Because of this, the partial differential expressions with respect to x has to change.

For example, initially: diff(w(x,y,t),x)=diff(w(x,y,t),x)
Later on, when the system gets modified in the presence of boundary layers on application of chain rule: diff(w(x,y,t,z),x)=diff(w(x,y,t,z),x)+diff(w(x,y,t,z),z)*diff(z,x)

This change has to happen for all higher order derivatives of x. In my case, I have upto 4th order derivative in x. How do I achieve these changes in the expressions in Maple in an elegant manner?

Good day everyone. I need to calculate series of the function given above. But I can not find result. It writes "Error, (in series/function) improper op or subscript selector". How can I fix this error? Please help me.

The detailed file has been attached above.
 

Ek4:=sqrt((Ea^2*Eb^2-Eab^2*X5^2)*Ds);

((((1/2)*k((1/2)*k^2/m2+X2)^2/m1((1/2)*k^2/m2+X2)+X1((1/2)*k^2/m2+X2))*((1/2)*k((1/2)*k^2/m2+X1)^2/m2((1/2)*k^2/m2+X1)+X2((1/2)*k^2/m2+X1))-((1/2)*k((1/2)*k^2/m2+X1)^2/m1((1/2)*k^2/m2+X1)+X2((1/2)*k^2/m2+X1))*X5^2)*(((1/2)*k((1/2)*k^2/m2+X2)^2/m1((1/2)*k^2/m2+X2)+X1((1/2)*k^2/m2+X2)-(1/2)*k((1/2)*k^2/m2+X1)^2/m2((1/2)*k^2/m2+X1)-X2((1/2)*k^2/m2+X1))^2+4*((1/2)*k((1/2)*k^2/m2+X1)^2/m1((1/2)*k^2/m2+X1)+X2((1/2)*k^2/m2+X1))^(1/2)*X5^2))^(1/2)

(5)

series(Ek4,k=0,4);

Error, (in series/function) improper op or subscript selector

 

series(sqrt(Ek4),k=0,4);

Error, (in series/function) improper op or subscript selector

 


 

Download function.mwfunction.mw

Hello,

I encounter the same error for upgrading the physics package

which was never solved for Maple 2020 !

See the screenshot.

This is again a problem of permission.

Best

I want to create a random polynomial using randpoly() with coefficients in a given list A := [1,1,2,5,6].

For that, I need to create a function that works as rand(1..9) but that when called returns a random element of A instead of a random number between 1 and 9.

What I have tried so far is

r := rand(1..numelems(A))
A[r()]

However, that returns a random element of A, not a function that returns a random element of A.

I've also tried

rand(A(1..numelems(A)))

But that gives an error.

One way of solving this problem would be to find out how to create a `..` type with the desired elements, something that works as 1..9 but with the elements of A. But convert(A,`..`) doesn't work and the rest of the things I tried don't work either.

Hi,

 I can’t use my "Filled" command to simultaneously explore my 3 graphs

Thanks for ideas

PétalesGraphes.mw

Hi, I am new to MAPLE and trying to solve the given BVP using shooting method but after defining the expression in u0 and u1, the code didn't compute the numerical value, and so that the whole code won't run exactly.
Please help me out to figure out the problem. I am looking for the answer. (NOTE: I copied this code from the web and it run so fine with the example but when I changed it according to me, it give me errors.)
The code is given below:
Thanks


restart;
with(DEtools);
with(plots);
dpdexi := 2;
K := 0.2;
epsilon := 0.0001;
                       epsilon := 0.0001
ode1 := (diff(phi(eta), eta) + epsilon*K*diff(phi(eta), eta)^3)/(1 + epsilon*diff(phi(eta), eta)^2) = dpdexi*eta;
bc1 := phi(0) = 0, phi(1) = 0;
a := 0;
b := 1;
ic1 := bc1[1], D(phi)(a) = alpha;
constraint := lhs(bc1[2]) - rhs(bc1[2]);
constraint = 0;
a0 := 0;
u0 := eval(constraint, phi(b) = PHIb(eval({ic1, ode1}, alpha = a0), phi(eta), b));
a1 := 0.01;
u1 := eval(constraint, phi(b) = PHIb(eval({ic1, ode1}, alpha = a1), phi(eta), b));
while fnormal(u1 - u0) <> 0 do
    z := solve(u1 = (u1 - u0)*(a1 - x)/(a1 - a0), x);
    a0, a1 := a1, z;
    u0, u1 := u1, eval(constraint, u(b) = Ub(eval({ic1, ode1}, alpha = z), u(x), b));
end do;
alpha_opt1 := z;
shoot_sol1 := evalf(dsolve(eval({ic1, ode1}, alpha = alpha_opt1), u(x), implicit));
infolevel[dsolve] := 3;
exact_sol1 := combine(dsolve({bc1, ode1}, u(x)));
infolevel[dsolve] := 0;
plot([rhs(exact_sol1), rhs(shoot_sol1)], x = 0 .. 1, color = [BLACK, RED], thickness = [5, 2], legend = ['exact', 'shooting'], title = "Figure 17.3");
evalf(eval(rhs(shoot_sol1), x = 1));

Hi..

OK, so my real-life example is I have a 3m gap between the house and the garage which I want to hang a sunshade. 
The hanging point on the house is 2.6m high and on the garage 2.1m (above ground). 

I have a choice between 2 sun sails, and I want to know -approximately- what sag I can expect from each.

I realise this is not exactly the same as the hanging wire problem i posted before.. but it my attempt anyway. Can anyone improve?

Sails.mw

there is this ref

https://www.mapleprimes.com/questions/147809-Real-Solution-For-Catenary-Differential-Equation

 

hello how do i define new units of multiple units. so i am able to choose them inside the Choose Unit in the rigth side

because as you can see there are not an option to define this as Unit -> [J/(K*mol)]

instead i have the option to use [m^2*kg/(s^2*K*mol)]

because i seen that you can defijne and install new units i just dont know how to impliment it so they alway remember my new units without i am suposed to initialize it with a command at top of my script

ps i also dont know how to define my new unit an just run that command at top of my script either.

but it would be better if it is permanent installed

i ask because i have several other units that is trublesome in maple (being in chemistry, we dont always use SI units) so molar [M] is also some times trublesome

 

as mapleprime often ask if we can include code i will do that

write this formula

3/(2)*8.3144621*J*K^(-1)*mol^(-1)

and use the simplify command in the right side and see that the unit is

thx

Hi. following on from

https://www.mapleprimes.com/questions/232817-Solving-Hanging-Cable-#comment281577

Second case: unequal poles.

I tried to work with vv's solution, but I got a problem... the required formula is

y=10.85378553130*cosh(0.0921337534371039*x) - 10.85378553130

Uneven.mw

Hi,

I have been trying to duplicate a solution to Schrodinger Eq from a utube video...the presenter use Wolfram software to graph and 

animate the plot...I have working on this all day..I a new user (several months)..any help would be appreciated.

I am attaching a screenshot

Thanks 

Frank

 

I am attaching my Maple worksheet for reference 
 

NULL

BELOW IS ADRESSING PARTICLE MOTION AND HOW ELECTRONS MOVE AROUND AND HOW MAGNETIC FIELDS AFFECTION THE MOTION OF ELECTRONS

 

PARTICLE MOTION "IGNORING SPIN" WILL COME BACK TO THAT

restart

with(Physics); interface(imaginaryunit = i)

Setup(hermitianoperators = {H, O__H, O__S}, realobjects = {`&hbar;`, m, t, x_}, combinepowersofsamebase = true, mathematicalnotation = true)

[combinepowersofsamebase = true, hermitianoperators = {H, O__H, O__S}, mathematicalnotation = true, realobjects = {`&hbar;`, m, t, x_}]

(1)

with(Physics[Vectors])

WAVE FUNCTION OF PARITCLE. t =time , x= some position vector

x_ = `<,>`(x, y, z)

x_ = Vector[column](%id = 36893489722226370188)

(2)

psi(t, x_)

psi(t, x_)

(3)

psi(t, x, y, z)

psi(t, x, y, z)

(4)

p_

p_

(5)

ψ is xome complex number with a real and imaginary number, the absolute value of abs(psi(t, x_))^2
`and`((is*the*probabilty*density*of*finding*the)*particle*at*postion*vector*x, at*time*t), ASSUME*SOME*POSITION*VECTOR*x, with*an*infintesmial*cube*around*it, infintesimal*volume*dV, PROBABILY*DENSITY*IS*abs(psi(t, x_))^2*dV, int(abs(psi(t, x_))^2, V) = 1

abs(psi(t, x_))^2

 

is*the^2*probabilty*density*of*finding*particle*at*postion*vector*x and at*time*t, ASSUME*SOME*POSITION*VECTOR*x, with*an*infintesmial*cube*around*it, infintesimal*volume*dV, PROBABILY*DENSITY*IS*abs(psi(t, x_))^2*dV, abs(psi(t, x_))^2*V = 1

(6)

THE ABOVE EQUATION IS TRUE IF THE WAVE FUNCTION IS "PROPERLY NORMALIZED"

THE WAVE FUNCTION EVOLVES IN TIME ACCORDING TO SCHRODINGER EQ

"`i&hbar;`(&PartialD;Psi)/(&PartialD;t)= -(`&hbar;`^(2))/(2 m)(((&PartialD;)^2)/((&PartialD;)^( )x^2)(psi)+((&PartialD;)^2)/((&PartialD;)^( )y^2)(psi)+((&PartialD;)^2)/((&PartialD;)^( )z^2)(psi))",
diff(psi, x, x)+diff(psi, y, y)+((diff(psi, z, z))*IS*THE*KENETIC*ENERGY*OF*THE)*PARTICLE

restart

psi(t, x, y, z)

psi(t, x, y, z)

(7)

Loading PDEtools

I*`&hbar;`*(diff(Ket(psi, t), t))

I*`&hbar;`*(diff(Ket(psi, t), t))

(8)

SElns := I*`&hbar;`*(diff(Ket(psi, t), t)); 'SElns'

SElns

(9)

SErhs1 := `&hbar;`^2*(diff(psi, t, x, x)+diff(psi, t, y, y)+diff(psi, t, z, z))/(2*m)

0

(10)

``

 

``

Classical physics example

KE = p^2/(2*m)

KE = (1/2)*p^2/m

(11)

KE = (p[x]^2+p[y]^2+p[z]^2)/(2*m)

KE = (1/2)*(p[x]^2+p[y]^2+p[z]^2)/m

(12)

QUANTUM MECHANICS

``

p[x] = `&hbar;`*(Diff(p[x], x))/(I)

p[x] = -I*`&hbar;`*(Diff(p[x], x))

(13)

 

p[x]^2 = Diff(rhs(p[x] = -I*`&hbar;`*(Diff(p[x], x))))

p[x]^2 = Diff(-I*`&hbar;`*(Diff(p[x], x)))

(14)

substituting each value of p[x], p[y],p[z]  IN EQ 11 is the KE

ONE VERY IMPORTANT SOLUTUION TO SE IS LISTED BELOW

psi(t, `#mover(mi("x"),mo("&rarr;"))`)= "Pi^(-3/(4)) sigma^(-3/(2))C(t)^(-3/(2))Exp((-1/(2))(((x)^(2))/(sigma^(2 )C(t)))+ ((i p *x-(i( p)^(2)(t/(`&hbar;`^())))/(2 m))/(C(t)))"

C(t) = 1 +  "(`i&hbar;` t))/((m sigma^(2))), sigma= initial position uncertainty of the particle,m,mass of particle"

```#mover(mi("p"),mo("&rarr;"))`=<p[x],p[y],p[z]>...just 3 Real no...this is NOT THE SAME AS EQ 14

There are 3 parmeters in the above eq σ (NOT PAULI), m ,`#mover(mi("p"),mo("&rarr;"))`

abs(psi(t, `#mover(mi("x"),mo("&rarr;"))`))^2 = 1/Pi^(3/2)Exp(-(`#mover(mi("x"),mo("&rarr;"))`-i*`#mover(mi("p"),mo("&rarr;"))`(t/m))^2/(sigma^2+(`#msup(mi("&hbar;"),mo("&InvisibleTimes;"))`*t/(sigma*m))^2))/(sigma^2+(`#msup(mi("&hbar;"),mo("&InvisibleTimes;"))`*t/(sigma*m))^2)^(3/2)

int(abs(psi(t, `#mover(mi("x",fontweight = "bold"),mo("&rarr;",fontweight = "bold"))`))^2, V) = 1

THE ABOVE FUNCTION WILL BE A MAXIMUM WHEN `#mover(mi("x"),mo("&rarr;"))` = `#mover(mi("p"),mo("&rarr;"))`*t/m

distribution,

n. 1. (Statistics) the set of possible values of a random variable, or points in a sample space , considered in terms of their theoretical or observed frequency . 2. also called generalized function. a generalization of the concept of a function, defined as continuous linear functionals  over spaces of infinitely differentiable functions, introduced so that all continuous functions possess partial distributional derivatives (also called Schwartzian derivatives) that are again distributions. This leads to so-called weak solutions of differential equations and is of importance in the theory of partial differential equations ...

THE EQ YIEDS A GAUSSIAN "BELL SHAPE IN 2 D ONLY, CANNOT BE DRAWN 3D, 
abs(psi)^2*y*axis, `#mover(mi("x"),mo("&rarr;"))` = `#mover(mi("p"),mo("&rarr;"))`*t*x*axis/m, width = sqrt(sigma^2+(`#msup(mi("&hbar;"),mo("&InvisibleTimes;"))`*t/(sigma*m))^2)*so*at*t and sqrt(sigma^2+(`#msup(mi("&hbar;"),mo("&InvisibleTimes;"))`*t/(sigma*m))^2)*so*at*t = ((0*this)*will*equal*sigma*initial*uncertainity*of*the*position*of)*the*particle

t→∞ width = `#msup(mi("&hbar;"),mo("&InvisibleTimes;"))`*t/(sigma*m)

``

restart

with(Student[VectorCalculus])

with(VectorCalculus)

with(plots)

 

ln(1) := ts

ts

(15)

ts := 2

2

(16)

sigma := 4

4

(17)

h := 1

1

(18)

M := 1

1

(19)

P := 1

1

(20)

x

x

(21)

psi(x, t)

psi(x, t)

(22)

I = sqrt(-1)

I = I

(23)

about(I)

I:
  All numeric values are properties as well as objects.
  Their location in the property lattice is obvious,
  in this case complex(extended_numeric).
 

 

h, I, ts, M

1, I, 2, 1

(24)

sigma^2

16

(25)

I*h*ts/M

2*I

(26)

 

about(I)

I:
  All numeric values are properties as well as objects.
  Their location in the property lattice is obvious,
  in this case complex(extended_numeric).
 

 

2+2*I

2+2*I

(27)

(-1/2)*(x^2)

-(1/2)*x^2

(28)

"((-1/2)((x^2)/((sigma^2+(I*h*(t/M))))"

NULL

-(1/2)*x^2/(2+2*I)

(-1/8+(1/8)*I)*x^2

(29)

``

``

"Exp(((-1/2)((x^2)/((sigma^2+(I*h*(t/M)))))+(((I *P* (x/h))- (I* ((P^2)/(2*M))*(t/h))/((1+(((I*h*t))/((M+sigma^2))))"

NULL

I*P*x/h

I*x

(30)

 

I*P^2*t/((2*M)*h)

((1/2)*I)*t

(31)

I*x-((1/2)*I)*t

I*x-((1/2)*I)*t

(32)

"((I *P* (x/h))- (I* ((P^2)/(2*M))*(t/h))/((1+(((I*h*t))/((M+sigma^2))))"

``

NULL

I*h*t

I*t

(33)

sigma^2+M

17

(34)

1+I*t*(1/17)

1+((1/17)*I)*t

(35)

(I*x-((1/2)*I)*t)/(1+((1/17)*I)*t)

(I*x-((1/2)*I)*t)/(1+((1/17)*I)*t)

(36)

NULL

exp(I*x-((1/2)*I)*t+(I*x-((1/2)*I)*t)/(1+((1/17)*I)*t))

exp(I*x-((1/2)*I)*t+(I*x-((1/2)*I)*t)/(1+((1/17)*I)*t))

(37)

psi(x, t) = exp(I*x-((1/2)*I)*t+(I*x-((1/2)*I)*t)/(1+((1/17)*I)*t))

psi(x, t) = exp(I*x-((1/2)*I)*t+(I*x-((1/2)*I)*t)/(1+((1/17)*I)*t))

(38)

simplify(psi(x, t) = exp(I*x-((1/2)*I)*t+(I*x-((1/2)*I)*t)/(1+((1/17)*I)*t)))

psi(x, t) = exp(-I*(-2*x+t)*(I*t+34)/((2*I)*t+34))

 

Explore(psi(x, t) = exp(VectorCalculus:-`-`(VectorCalculus:-`*`(VectorCalculus:-`*`(VectorCalculus:-`*`(I, `-`(2*x)+t), VectorCalculus:-`+`(VectorCalculus:-`*`(I, t), 34)), 1/VectorCalculus:-`+`(VectorCalculus:-`*`(VectorCalculus:-`*`(2, I), t), 34)))), parameters = [[t = 0 .. 40, controller = slider], [x = -20 .. 80, controller = slider]], loop = never, size = NoUserValue, numeric = false, echoexpression = true)

(39)

with(plots)

implicitplot(abs(psi(x, ts*t)), Re(psi(x, ts*t)), Im(psi(x, ts*t)), x = -20 .. 80, scaling = constrained)

Error, (in plots/implicitplot) invalid input: lhs received Im(psi(x, 2*t)), which is not valid for its 1st argument, expr

 

NULL


 

Download Lec7QuantumWaveSE.mw
 

NULL

BELOW IS ADRESSING PARTICLE MOTION AND HOW ELECTRONS MOVE AROUND AND HOW MAGNETIC FIELDS AFFECTION THE MOTION OF ELECTRONS

 

PARTICLE MOTION "IGNORING SPIN" WILL COME BACK TO THAT

restart

with(Physics); interface(imaginaryunit = i)

Setup(hermitianoperators = {H, O__H, O__S}, realobjects = {`&hbar;`, m, t, x_}, combinepowersofsamebase = true, mathematicalnotation = true)

[combinepowersofsamebase = true, hermitianoperators = {H, O__H, O__S}, mathematicalnotation = true, realobjects = {`&hbar;`, m, t, x_}]

(1)

with(Physics[Vectors])

WAVE FUNCTION OF PARITCLE. t =time , x= some position vector

x_ = `<,>`(x, y, z)

x_ = Vector[column](%id = 36893489722226370188)

(2)

psi(t, x_)

psi(t, x_)

(3)

psi(t, x, y, z)

psi(t, x, y, z)

(4)

p_

p_

(5)

ψ is xome complex number with a real and imaginary number, the absolute value of abs(psi(t, x_))^2
`and`((is*the*probabilty*density*of*finding*the)*particle*at*postion*vector*x, at*time*t), ASSUME*SOME*POSITION*VECTOR*x, with*an*infintesmial*cube*around*it, infintesimal*volume*dV, PROBABILY*DENSITY*IS*abs(psi(t, x_))^2*dV, int(abs(psi(t, x_))^2, V) = 1

abs(psi(t, x_))^2

 

is*the^2*probabilty*density*of*finding*particle*at*postion*vector*x and at*time*t, ASSUME*SOME*POSITION*VECTOR*x, with*an*infintesmial*cube*around*it, infintesimal*volume*dV, PROBABILY*DENSITY*IS*abs(psi(t, x_))^2*dV, abs(psi(t, x_))^2*V = 1

(6)

THE ABOVE EQUATION IS TRUE IF THE WAVE FUNCTION IS "PROPERLY NORMALIZED"

THE WAVE FUNCTION EVOLVES IN TIME ACCORDING TO SCHRODINGER EQ

"`i&hbar;`(&PartialD;Psi)/(&PartialD;t)= -(`&hbar;`^(2))/(2 m)(((&PartialD;)^2)/((&PartialD;)^( )x^2)(psi)+((&PartialD;)^2)/((&PartialD;)^( )y^2)(psi)+((&PartialD;)^2)/((&PartialD;)^( )z^2)(psi))",
diff(psi, x, x)+diff(psi, y, y)+((diff(psi, z, z))*IS*THE*KENETIC*ENERGY*OF*THE)*PARTICLE

restart

psi(t, x, y, z)

psi(t, x, y, z)

(7)

Loading PDEtools

I*`&hbar;`*(diff(Ket(psi, t), t))

I*`&hbar;`*(diff(Ket(psi, t), t))

(8)

SElns := I*`&hbar;`*(diff(Ket(psi, t), t)); 'SElns'

SElns

(9)

SErhs1 := `&hbar;`^2*(diff(psi, t, x, x)+diff(psi, t, y, y)+diff(psi, t, z, z))/(2*m)

0

(10)

``

 

``

Classical physics example

KE = p^2/(2*m)

KE = (1/2)*p^2/m

(11)

KE = (p[x]^2+p[y]^2+p[z]^2)/(2*m)

KE = (1/2)*(p[x]^2+p[y]^2+p[z]^2)/m

(12)

QUANTUM MECHANICS

``

p[x] = `&hbar;`*(Diff(p[x], x))/(I)

p[x] = -I*`&hbar;`*(Diff(p[x], x))

(13)

 

p[x]^2 = Diff(rhs(p[x] = -I*`&hbar;`*(Diff(p[x], x))))

p[x]^2 = Diff(-I*`&hbar;`*(Diff(p[x], x)))

(14)

substituting each value of p[x], p[y],p[z]  IN EQ 11 is the KE

ONE VERY IMPORTANT SOLUTUION TO SE IS LISTED BELOW

psi(t, `#mover(mi("x"),mo("&rarr;"))`)= "Pi^(-3/(4)) sigma^(-3/(2))C(t)^(-3/(2))Exp((-1/(2))(((x)^(2))/(sigma^(2 )C(t)))+ ((i p *x-(i( p)^(2)(t/(`&hbar;`^())))/(2 m))/(C(t)))"

C(t) = 1 +  "(`i&hbar;` t))/((m sigma^(2))), sigma= initial position uncertainty of the particle,m,mass of particle"

```#mover(mi("p"),mo("&rarr;"))`=<p[x],p[y],p[z]>...just 3 Real no...this is NOT THE SAME AS EQ 14

There are 3 parmeters in the above eq σ (NOT PAULI), m ,`#mover(mi("p"),mo("&rarr;"))`

abs(psi(t, `#mover(mi("x"),mo("&rarr;"))`))^2 = 1/Pi^(3/2)Exp(-(`#mover(mi("x"),mo("&rarr;"))`-i*`#mover(mi("p"),mo("&rarr;"))`(t/m))^2/(sigma^2+(`#msup(mi("&hbar;"),mo("&InvisibleTimes;"))`*t/(sigma*m))^2))/(sigma^2+(`#msup(mi("&hbar;"),mo("&InvisibleTimes;"))`*t/(sigma*m))^2)^(3/2)

int(abs(psi(t, `#mover(mi("x",fontweight = "bold"),mo("&rarr;",fontweight = "bold"))`))^2, V) = 1

THE ABOVE FUNCTION WILL BE A MAXIMUM WHEN `#mover(mi("x"),mo("&rarr;"))` = `#mover(mi("p"),mo("&rarr;"))`*t/m

distribution,

n. 1. (Statistics) the set of possible values of a random variable, or points in a sample space , considered in terms of their theoretical or observed frequency . 2. also called generalized function. a generalization of the concept of a function, defined as continuous linear functionals  over spaces of infinitely differentiable functions, introduced so that all continuous functions possess partial distributional derivatives (also called Schwartzian derivatives) that are again distributions. This leads to so-called weak solutions of differential equations and is of importance in the theory of partial differential equations ...

THE EQ YIEDS A GAUSSIAN "BELL SHAPE IN 2 D ONLY, CANNOT BE DRAWN 3D, 
abs(psi)^2*y*axis, `#mover(mi("x"),mo("&rarr;"))` = `#mover(mi("p"),mo("&rarr;"))`*t*x*axis/m, width = sqrt(sigma^2+(`#msup(mi("&hbar;"),mo("&InvisibleTimes;"))`*t/(sigma*m))^2)*so*at*t and sqrt(sigma^2+(`#msup(mi("&hbar;"),mo("&InvisibleTimes;"))`*t/(sigma*m))^2)*so*at*t = ((0*this)*will*equal*sigma*initial*uncertainity*of*the*position*of)*the*particle

t→∞ width = `#msup(mi("&hbar;"),mo("&InvisibleTimes;"))`*t/(sigma*m)

``

restart

with(Student[VectorCalculus])

with(VectorCalculus)

with(plots)

 

ln(1) := ts

ts

(15)

ts := 2

2

(16)

sigma := 4

4

(17)

h := 1

1

(18)

M := 1

1

(19)

P := 1

1

(20)

x

x

(21)

psi(x, t)

psi(x, t)

(22)

I = sqrt(-1)

I = I

(23)

about(I)

I:
  All numeric values are properties as well as objects.
  Their location in the property lattice is obvious,
  in this case complex(extended_numeric).
 

 

h, I, ts, M

1, I, 2, 1

(24)

sigma^2

16

(25)

I*h*ts/M

2*I

(26)

 

about(I)

I:
  All numeric values are properties as well as objects.
  Their location in the property lattice is obvious,
  in this case complex(extended_numeric).
 

 

2+2*I

2+2*I

(27)

(-1/2)*(x^2)

-(1/2)*x^2

(28)

"((-1/2)((x^2)/((sigma^2+(I*h*(t/M))))"

NULL

-(1/2)*x^2/(2+2*I)

(-1/8+(1/8)*I)*x^2

(29)

``

``

"Exp(((-1/2)((x^2)/((sigma^2+(I*h*(t/M)))))+(((I *P* (x/h))- (I* ((P^2)/(2*M))*(t/h))/((1+(((I*h*t))/((M+sigma^2))))"

NULL

I*P*x/h

I*x

(30)

 

I*P^2*t/((2*M)*h)

((1/2)*I)*t

(31)

I*x-((1/2)*I)*t

I*x-((1/2)*I)*t

(32)

"((I *P* (x/h))- (I* ((P^2)/(2*M))*(t/h))/((1+(((I*h*t))/((M+sigma^2))))"

``

NULL

I*h*t

I*t

(33)

sigma^2+M

17

(34)

1+I*t*(1/17)

1+((1/17)*I)*t

(35)

(I*x-((1/2)*I)*t)/(1+((1/17)*I)*t)

(I*x-((1/2)*I)*t)/(1+((1/17)*I)*t)

(36)

NULL

exp(I*x-((1/2)*I)*t+(I*x-((1/2)*I)*t)/(1+((1/17)*I)*t))

exp(I*x-((1/2)*I)*t+(I*x-((1/2)*I)*t)/(1+((1/17)*I)*t))

(37)

psi(x, t) = exp(I*x-((1/2)*I)*t+(I*x-((1/2)*I)*t)/(1+((1/17)*I)*t))

psi(x, t) = exp(I*x-((1/2)*I)*t+(I*x-((1/2)*I)*t)/(1+((1/17)*I)*t))

(38)

simplify(psi(x, t) = exp(I*x-((1/2)*I)*t+(I*x-((1/2)*I)*t)/(1+((1/17)*I)*t)))

psi(x, t) = exp(-I*(-2*x+t)*(I*t+34)/((2*I)*t+34))

 

Explore(psi(x, t) = exp(VectorCalculus:-`-`(VectorCalculus:-`*`(VectorCalculus:-`*`(VectorCalculus:-`*`(I, `-`(2*x)+t), VectorCalculus:-`+`(VectorCalculus:-`*`(I, t), 34)), 1/VectorCalculus:-`+`(VectorCalculus:-`*`(VectorCalculus:-`*`(2, I), t), 34)))), parameters = [[t = 0 .. 40, controller = slider], [x = -20 .. 80, controller = slider]], loop = never, size = NoUserValue, numeric = false, echoexpression = true)

(39)

with(plots)

implicitplot(abs(psi(x, ts*t)), Re(psi(x, ts*t)), Im(psi(x, ts*t)), x = -20 .. 80, scaling = constrained)

Error, (in plots/implicitplot) invalid input: lhs received Im(psi(x, 2*t)), which is not valid for its 1st argument, expr

 

NULL


 

Download Lec7QuantumWaveSE.mw
 

NULL

BELOW IS ADRESSING PARTICLE MOTION AND HOW ELECTRONS MOVE AROUND AND HOW MAGNETIC FIELDS AFFECTION THE MOTION OF ELECTRONS

 

PARTICLE MOTION "IGNORING SPIN" WILL COME BACK TO THAT

restart

with(Physics); interface(imaginaryunit = i)

Setup(hermitianoperators = {H, O__H, O__S}, realobjects = {`&hbar;`, m, t, x_}, combinepowersofsamebase = true, mathematicalnotation = true)

[combinepowersofsamebase = true, hermitianoperators = {H, O__H, O__S}, mathematicalnotation = true, realobjects = {`&hbar;`, m, t, x_}]

(1)

with(Physics[Vectors])

WAVE FUNCTION OF PARITCLE. t =time , x= some position vector

x_ = `<,>`(x, y, z)

x_ = Vector[column](%id = 36893489722226370188)

(2)

psi(t, x_)

psi(t, x_)

(3)

psi(t, x, y, z)

psi(t, x, y, z)

(4)

p_

p_

(5)

ψ is xome complex number with a real and imaginary number, the absolute value of abs(psi(t, x_))^2
`and`((is*the*probabilty*density*of*finding*the)*particle*at*postion*vector*x, at*time*t), ASSUME*SOME*POSITION*VECTOR*x, with*an*infintesmial*cube*around*it, infintesimal*volume*dV, PROBABILY*DENSITY*IS*abs(psi(t, x_))^2*dV, int(abs(psi(t, x_))^2, V) = 1

abs(psi(t, x_))^2

 

is*the^2*probabilty*density*of*finding*particle*at*postion*vector*x and at*time*t, ASSUME*SOME*POSITION*VECTOR*x, with*an*infintesmial*cube*around*it, infintesimal*volume*dV, PROBABILY*DENSITY*IS*abs(psi(t, x_))^2*dV, abs(psi(t, x_))^2*V = 1

(6)

THE ABOVE EQUATION IS TRUE IF THE WAVE FUNCTION IS "PROPERLY NORMALIZED"

THE WAVE FUNCTION EVOLVES IN TIME ACCORDING TO SCHRODINGER EQ

"`i&hbar;`(&PartialD;Psi)/(&PartialD;t)= -(`&hbar;`^(2))/(2 m)(((&PartialD;)^2)/((&PartialD;)^( )x^2)(psi)+((&PartialD;)^2)/((&PartialD;)^( )y^2)(psi)+((&PartialD;)^2)/((&PartialD;)^( )z^2)(psi))",
diff(psi, x, x)+diff(psi, y, y)+((diff(psi, z, z))*IS*THE*KENETIC*ENERGY*OF*THE)*PARTICLE

restart

psi(t, x, y, z)

psi(t, x, y, z)

(7)

Loading PDEtools

I*`&hbar;`*(diff(Ket(psi, t), t))

I*`&hbar;`*(diff(Ket(psi, t), t))

(8)

SElns := I*`&hbar;`*(diff(Ket(psi, t), t)); 'SElns'

SElns

(9)

SErhs1 := `&hbar;`^2*(diff(psi, t, x, x)+diff(psi, t, y, y)+diff(psi, t, z, z))/(2*m)

0

(10)

``

 

``

Classical physics example

KE = p^2/(2*m)

KE = (1/2)*p^2/m

(11)

KE = (p[x]^2+p[y]^2+p[z]^2)/(2*m)

KE = (1/2)*(p[x]^2+p[y]^2+p[z]^2)/m

(12)

QUANTUM MECHANICS

``

p[x] = `&hbar;`*(Diff(p[x], x))/(I)

p[x] = -I*`&hbar;`*(Diff(p[x], x))

(13)

 

p[x]^2 = Diff(rhs(p[x] = -I*`&hbar;`*(Diff(p[x], x))))

p[x]^2 = Diff(-I*`&hbar;`*(Diff(p[x], x)))

(14)

substituting each value of p[x], p[y],p[z]  IN EQ 11 is the KE

ONE VERY IMPORTANT SOLUTUION TO SE IS LISTED BELOW

psi(t, `#mover(mi("x"),mo("&rarr;"))`)= "Pi^(-3/(4)) sigma^(-3/(2))C(t)^(-3/(2))Exp((-1/(2))(((x)^(2))/(sigma^(2 )C(t)))+ ((i p *x-(i( p)^(2)(t/(`&hbar;`^())))/(2 m))/(C(t)))"

C(t) = 1 +  "(`i&hbar;` t))/((m sigma^(2))), sigma= initial position uncertainty of the particle,m,mass of particle"

```#mover(mi("p"),mo("&rarr;"))`=<p[x],p[y],p[z]>...just 3 Real no...this is NOT THE SAME AS EQ 14

There are 3 parmeters in the above eq σ (NOT PAULI), m ,`#mover(mi("p"),mo("&rarr;"))`

abs(psi(t, `#mover(mi("x"),mo("&rarr;"))`))^2 = 1/Pi^(3/2)Exp(-(`#mover(mi("x"),mo("&rarr;"))`-i*`#mover(mi("p"),mo("&rarr;"))`(t/m))^2/(sigma^2+(`#msup(mi("&hbar;"),mo("&InvisibleTimes;"))`*t/(sigma*m))^2))/(sigma^2+(`#msup(mi("&hbar;"),mo("&InvisibleTimes;"))`*t/(sigma*m))^2)^(3/2)

int(abs(psi(t, `#mover(mi("x",fontweight = "bold"),mo("&rarr;",fontweight = "bold"))`))^2, V) = 1

THE ABOVE FUNCTION WILL BE A MAXIMUM WHEN `#mover(mi("x"),mo("&rarr;"))` = `#mover(mi("p"),mo("&rarr;"))`*t/m

distribution,

n. 1. (Statistics) the set of possible values of a random variable, or points in a sample space , considered in terms of their theoretical or observed frequency . 2. also called generalized function. a generalization of the concept of a function, defined as continuous linear functionals  over spaces of infinitely differentiable functions, introduced so that all continuous functions possess partial distributional derivatives (also called Schwartzian derivatives) that are again distributions. This leads to so-called weak solutions of differential equations and is of importance in the theory of partial differential equations ...

THE EQ YIEDS A GAUSSIAN "BELL SHAPE IN 2 D ONLY, CANNOT BE DRAWN 3D, 
abs(psi)^2*y*axis, `#mover(mi("x"),mo("&rarr;"))` = `#mover(mi("p"),mo("&rarr;"))`*t*x*axis/m, width = sqrt(sigma^2+(`#msup(mi("&hbar;"),mo("&InvisibleTimes;"))`*t/(sigma*m))^2)*so*at*t and sqrt(sigma^2+(`#msup(mi("&hbar;"),mo("&InvisibleTimes;"))`*t/(sigma*m))^2)*so*at*t = ((0*this)*will*equal*sigma*initial*uncertainity*of*the*position*of)*the*particle

t→∞ width = `#msup(mi("&hbar;"),mo("&InvisibleTimes;"))`*t/(sigma*m)

``

restart

with(Student[VectorCalculus])

with(VectorCalculus)

with(plots)

 

ln(1) := ts

ts

(15)

ts := 2

2

(16)

sigma := 4

4

(17)

h := 1

1

(18)

M := 1

1

(19)

P := 1

1

(20)

x

x

(21)

psi(x, t)

psi(x, t)

(22)

I = sqrt(-1)

I = I

(23)

about(I)

I:
  All numeric values are properties as well as objects.
  Their location in the property lattice is obvious,
  in this case complex(extended_numeric).
 

 

h, I, ts, M

1, I, 2, 1

(24)

sigma^2

16

(25)

I*h*ts/M

2*I

(26)

 

about(I)

I:
  All numeric values are properties as well as objects.
  Their location in the property lattice is obvious,
  in this case complex(extended_numeric).
 

 

2+2*I

2+2*I

(27)

(-1/2)*(x^2)

-(1/2)*x^2

(28)

"((-1/2)((x^2)/((sigma^2+(I*h*(t/M))))"

NULL

-(1/2)*x^2/(2+2*I)

(-1/8+(1/8)*I)*x^2

(29)

``

``

"Exp(((-1/2)((x^2)/((sigma^2+(I*h*(t/M)))))+(((I *P* (x/h))- (I* ((P^2)/(2*M))*(t/h))/((1+(((I*h*t))/((M+sigma^2))))"

NULL

I*P*x/h

I*x

(30)

 

I*P^2*t/((2*M)*h)

((1/2)*I)*t

(31)

I*x-((1/2)*I)*t

I*x-((1/2)*I)*t

(32)

"((I *P* (x/h))- (I* ((P^2)/(2*M))*(t/h))/((1+(((I*h*t))/((M+sigma^2))))"

``

NULL

I*h*t

I*t

(33)

sigma^2+M

17

(34)

1+I*t*(1/17)

1+((1/17)*I)*t

(35)

(I*x-((1/2)*I)*t)/(1+((1/17)*I)*t)

(I*x-((1/2)*I)*t)/(1+((1/17)*I)*t)

(36)

NULL

exp(I*x-((1/2)*I)*t+(I*x-((1/2)*I)*t)/(1+((1/17)*I)*t))

exp(I*x-((1/2)*I)*t+(I*x-((1/2)*I)*t)/(1+((1/17)*I)*t))

(37)

psi(x, t) = exp(I*x-((1/2)*I)*t+(I*x-((1/2)*I)*t)/(1+((1/17)*I)*t))

psi(x, t) = exp(I*x-((1/2)*I)*t+(I*x-((1/2)*I)*t)/(1+((1/17)*I)*t))

(38)

simplify(psi(x, t) = exp(I*x-((1/2)*I)*t+(I*x-((1/2)*I)*t)/(1+((1/17)*I)*t)))

psi(x, t) = exp(-I*(-2*x+t)*(I*t+34)/((2*I)*t+34))

 

Explore(psi(x, t) = exp(VectorCalculus:-`-`(VectorCalculus:-`*`(VectorCalculus:-`*`(VectorCalculus:-`*`(I, `-`(2*x)+t), VectorCalculus:-`+`(VectorCalculus:-`*`(I, t), 34)), 1/VectorCalculus:-`+`(VectorCalculus:-`*`(VectorCalculus:-`*`(2, I), t), 34)))), parameters = [[t = 0 .. 40, controller = slider], [x = -20 .. 80, controller = slider]], loop = never, size = NoUserValue, numeric = false, echoexpression = true)

(39)

with(plots)

implicitplot(abs(psi(x, ts*t)), Re(psi(x, ts*t)), Im(psi(x, ts*t)), x = -20 .. 80, scaling = constrained)

Error, (in plots/implicitplot) invalid input: lhs received Im(psi(x, 2*t)), which is not valid for its 1st argument, expr

 

NULL


 

Download Lec7QuantumWaveSE.mw

 

 

I am trying to plot a Runge-Kutta method for 1+ tsin(tx). When ever I try to do the plot data command an empty graph shows up. I noticed that some of my values from the algorithm did not calculate properly. How do I fix this problem? This is what I typed in for the calculations. 

f := (t, x) -> 1 + t*sin(x);
t[0] := 0;x[0] := 0;
h := 0.1;
 

for n to 20 do
    t[n] := n*h;
    m1 := f(t[n - 1], x[n - 1]);
    m2 := f(t[n - 1] + h/2, x[n - 1] + m1*h/2);
    m3 := f(t[n - 1] + h/2, x[n - 1] + m2*h/2);
    m4 := f(t[n - 1] + h, h*m3 + x[n - 1]);
    x[n] := x[n - 1] + h/6*(m1 + 2*m2 + 2*m3 + m4);
end;
 

Hi

I am trying to follow this paper

http://euclid.trentu.ca/aejm/V4N1/Chatterjee.V4N1.pdf

Lets start with the easier problem, equal poles. Assume that the length of the cable is 120m and the two poles have equal height of 50m. Our goal is to determine the minimum distance between the two poles that will prevent the cable from touching the ground.

I was trying to get Maple to agree with their derived formula, namely

y(x)= 11*cosh(1/11*x) - 61,

but I think I have not set the IC's correctly. or provided for the length of the cable.

restart:
int(sqrt(1+diff(y(t),t)^2),t=0..x)=120/2

//can't solve the above directly, or maybe someone clever here can

//a is a constant
DIV := diff(y(x), x, x) = a*sqrt(1 + diff(y(x), x)^2);
RV := y(0) = 0, D(y)(0) = -50;
dsolve({DIV});
Opl := dsolve({DIV, RV}, y(x));
allvalues(%)
 

 

Dear all,

Reversion of series---computing a series for the functional inverse of a function---has been in Maple since forever, but many people are not aware of how easy it is.  Here's an example, where we are looking for "self-reverting" series---which I called "ambiverts".  Anyway have fun.

 

https://maple.cloud/app/5974582695821312/Series+Reversion%3A+Looking+for+ambiverts

PS There looks to be some "code rot" in the branch point series for Lambert W in Maple, which we encounter in that worksheet.  Or, I may simply have not coded it very well in the first place (yeah, that was mine, once upon a time).  Checking now.  But there is a workaround (albeit an ugly one) shown in that worksheet.

 

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