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Kitonum

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17 years, 29 days
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These are answers submitted by Kitonum

Another way...

Kitonum 20084
September 05 2020
0 3

We easily get the same result (without the VectorCalculus package) using the standard way to calculate such a curvilinear integral:

restart;
x:=1+t: y:=2*t: # Parametric equations of the line with endpoints (1,0), (2,2)
int( (2*x + y^2)*sqrt(diff(x,t)^2+diff(y,t)^2), t=0..1);

                                        

Manually without Maple...

Kitonum 20084
September 02 2020
1 0

I think that from a pedagogical point of view, a student studying mathematical analysis should be able to solve such simple examples by hand, and only then, of course, can he check the correctness of the calculations using Maple:

     

 

combine, factor...

Kitonum 20084
September 01 2020
1 1


 

> 

restart;
subexp := M__a*sin(omega*t + alpha)*I__a*sin(omega*t + phi);
subexp2:=factor(combine(subexp));
d:=M__a*I__a:
subexp22:=d*``(subexp2/d)

 

M__a*sin(omega*t+alpha)*I__a*sin(omega*t+phi)

  

(1/2)*M__a*I__a*(cos(alpha-phi)-cos(2*omega*t+alpha+phi))

  

M__a*I__a*``((1/2)*cos(alpha-phi)-(1/2)*cos(2*omega*t+alpha+phi))

 (1)
> 

 


 

Download identity.mw

Solution...

Kitonum 20084
August 31 2020
0 1 
restart;
with(Statistics):
X:=Vector([0,0.001,0.002,0.003,0.004,0.005,0.006,0.007,0.008,0.009,0.01,0.012]):
Y:=Vector([1.103,1.057,1.016,0.978,0.94,0.91,0.88,0.85,0.826,0.8,0.778,0.735]):
Fn:=Fit((-a*t+b)/(-d+e*t+f*t^3),X,Y,t);
add((Y-~[seq(eval(Fn,t=x),x=X)])^~2); # residual sum of squares
plots:-display(plot(X,Y,style=point,color=red,symbolsize=12), plot(Fn,t=X[1]..X[-1],color=blue), labels=["X","Y"]);

Download RS.mw

solve...

Kitonum 20084
August 30 2020
2 0


 

> 

restart;
A:=x<7: B:=-10<x and x<25: C:=x>15:
> 

solve(A and B);

RealRange(Open(-10), 7)

 (1)
> 

solve(B or C);

RealRange(Open(-10), infinity)

 (2)
> 

solve(A and B and C); # Return NULL that is empty set
> 

solve(A or B or C); # Whole real number axis

x

 (3)
> 

 


 

Download solve.mw

A way...

Kitonum 20084
August 29 2020
1 2


 

> 

restart;
S:=proc(n)
local P;
uses combinat;
P:=permute([0$n,1$n]):
select(p->andmap(k->add(p[1..k])<=k/2, [$ 1..2*n]), P);
end proc:

S(3);
nops(%);
seq(nops(S(n)), n=1..7);

 

[[0, 0, 0, 1, 1, 1], [0, 0, 1, 0, 1, 1], [0, 0, 1, 1, 0, 1], [0, 1, 0, 0, 1, 1], [0, 1, 0, 1, 0, 1]]

  

5

  

1, 2, 5, 14, 42, 132, 429

 (1)
> 

 


Edit.

Download seq_edit.mw

Procedure...

Kitonum 20084
August 28 2020
1 4

To avoid repeated calls to Typesetting:-Settings , you can enter your equations through the procedure  P  below:

> 

restart;
P:=proc(ode)
local t;
t:=indets(ode, name)[];
Typesetting:-Settings(usedot=false,prime=t,typesetprime=true):
ode;
end proc:

ode1:=P(diff(y(t),t$2)+diff(y(t),t)+y(t)= 0);
ode2:=P(diff(y(x),x$2)+diff(y(x),x)+y(x)= 0);

diff(diff(y(t), t), t)+diff(y(t), t)+y(t) = 0

  

diff(diff(y(x), x), x)+diff(y(x), x)+y(x) = 0

 (1)
> 

 


 

Download prime.mw

assign...

Kitonum 20084
August 28 2020
2 0

Use  assign  for this instead of  :=  :

restart;
for a from 1 to 2 do
assign(convert(cat("k",a),symbol)=a);
od;
k1, k2;

                                             1, 2

Here is another way of assignment through the seq command (multiple assignment), where  :=  doesn't work, but  assign  does:

restart;
seq(k||i, i=1..2) := seq(a, a=1..2);  # an error
assign(seq(k||i=i, i=1..2));  # OK
k1, k2;  # check

   Error, invalid left hand side in assignment
                              1, 2

allsolutions option...

Kitonum 20084
August 26 2020
2 1


 

> 

restart;
assume(omega>0);assume(zeta>0 and zeta<1);
tf:=omega^2/(s^2+2*zeta*omega*s+omega^2);
y:=tf*1/s;
yt:=inttrans:-invlaplace(y,s,t);
dyt:=diff(yt,t);

solve(dyt=0,t, allsolutions);
about(_Z1);

omega^2/(2*omega*s*zeta+omega^2+s^2)

  

omega^2/((2*omega*s*zeta+omega^2+s^2)*s)

  

1-exp(-omega*zeta*t)*cos((-zeta^2+1)^(1/2)*omega*t)-zeta*exp(-omega*zeta*t)*sin((-zeta^2+1)^(1/2)*omega*t)/(-zeta^2+1)^(1/2)

  

exp(-omega*zeta*t)*(-zeta^2+1)^(1/2)*omega*sin((-zeta^2+1)^(1/2)*omega*t)+zeta^2*omega*exp(-omega*zeta*t)*sin((-zeta^2+1)^(1/2)*omega*t)/(-zeta^2+1)^(1/2)

  

Pi*_Z1/((-zeta^2+1)^(1/2)*omega)

  

Originally _Z1, renamed _Z1~:
  is assumed to be: integer

  
> 

 


 

Download dyt.mw

A way...

Kitonum 20084
August 23 2020
0 0

Let's define new functions  Sin, Cos, Tan, Cot  for which arguments are assumed in degrees:

Sin:=x->sin(x*Pi/180):
Cos:=x->cos(x*Pi/180):
Tan:=x->tan(x*Pi/180):
Cot:=x->cot(x*Pi/180):

# Examples
Sin(30);
Tan(45);
Cos(30);

 

Approximate calculation...

Kitonum 20084
August 19 2020
2 0

Probably this integral cannot be calculated symbolically (even for the specific values of the parameters). So calculate numerically.

Example:


 

> 

restart;
t:=1: A:=2: omega:=3: tau:=4: phi:=5:
int( ((-A*omega*sin(omega*x+phi)*exp(-x/tau) - A*cos(omega*x+phi)*exp(-x/tau)/tau)^2 + 1)^(1/2), x=0..t );
int( ((-A*omega*sin(omega*x+phi)*exp(-x/tau) - A*cos(omega*x+phi)*exp(-x/tau)/tau)^2 + 1)^(1/2), x=0..t, numeric );

int(((-6*sin(3*x+5)*exp(-(1/4)*x)-(1/2)*cos(3*x+5)*exp(-(1/4)*x))^2+1)^(1/2), x = 0 .. 1)

  

3.482744646

 (1)
> 

 


 

Download int.mw

Ways...

Kitonum 20084
August 19 2020
2 1

This can be done in many ways. Here are two options (long and short):


 

> 

restart;
data:=[0,12,0,7,5,3,7,10,0,0,9,3,2,5,0,6]:
N:=0:
for d in data do
if d>=7 then N:=N+1 fi;
od:
N;

5

 (1)
> 

nops(select(`>=`, data, 7));

5

 (2)
> 

 


 

Download 1.mw

match...

Kitonum 20084
August 17 2020
0 1


 

> 

restart;
eq1:= (-k*I + 2*I + m)*sqrt(3) - 3*I*m - 3*k;
eq2:=eq1/2;
eq_given:= a*(k + I*m) + b;
match(eq2 = eq_given, {k,m}, 's');
s;

(-I*k+2*I+m)*3^(1/2)-(3*I)*m-3*k

  

(1/2)*(-I*k+2*I+m)*3^(1/2)-((3/2)*I)*m-(3/2)*k

  

a*(k+I*m)+b

  

true

  

{a = -((1/2)*I)*(3^(1/2)-3*I), a = -((1/2)*I)*3^(1/2)-3/2, b = I*3^(1/2)}

 (1)
> 

 


 

Download Q20200817_new.mw

A way...

Kitonum 20084
August 17 2020
1 0


 

> 

restart;
Expr:=W__1 + W__2 = -sin(-beta + alpha)*((H^2 - h^2)*gamma + h^2*psi)/(2*sin(beta)*sin(alpha));
applyop(p->H^2*p,2,applyop(p->collect(expand(p/H^2),gamma),[2,3],Expr));

W__1+W__2 = -(1/2)*sin(-beta+alpha)*((H^2-h^2)*gamma+h^2*psi)/(sin(beta)*sin(alpha))

  

W__1+W__2 = -(1/2)*H^2*sin(-beta+alpha)*((1-h^2/H^2)*gamma+h^2*psi/H^2)/(sin(beta)*sin(alpha))

 (1)
> 

 


 

Download 1.mw

Another way...

Kitonum 20084
August 15 2020
1 1

Example:

restart;
N:=8: # The number of the points
plot([seq([n/(N-1),ln(1+sin(Pi*n/(N-1)))], n=0..N-1)], legend = numerical, style = point, symbol = box, color = blue, symbolsize = 15);

 

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