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Kitonum

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17 years, 49 days
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These are answers submitted by Kitonum

D...

Kitonum 21475
September 22 2020
1 0

Example:

restart;
f:=x->x^3;
(D@@2)(f)(x);
(D@@2)(f)(3);

 

The proof of 2) and 3)...

Kitonum 21475
September 22 2020
0 0

Let (x0,y0) is a point on the ellipse E.


 

> 

restart;
eq:=x0*x/a^2+y0*y/b^2=1: # The equation of the tangent in the point(x0,y0) (from wiki)
OT:=solve(eval(eq,y=0),x);
Om:=x0:
Om*OT;

k:=-coeff(lhs(eq),x)/coeff(lhs(eq),y); # The slope of the line eq
eq1:=y-y0=-1/k*(x-x0); # The equation of the normal in the point(x0,y0)
ON:=solve(eval(eq1,y=0),x);
simplify(ON,{a^2-b^2=c^2});
 

a^2/x0

  

a^2

  

-x0*b^2/(a^2*y0)

  

y-y0 = a^2*y0*(x-x0)/(x0*b^2)

  

(a^2-b^2)*x0/a^2

  

c^2*x0/a^2

 (1)
> 

 


Edit.

Download prop_23.mw

 

plottools:-reflect...

Kitonum 21475
September 22 2020
0 0

Or (an example):

restart;
p:=plot(exp(x), x=-2..2);
plottools:-reflect(p, [[0,0],[1,1]]);

 

Procedure for this...

Kitonum 21475
September 21 2020
1 0

Probably the procedures  IntegerPoints (or  IntegerPoints1) from the post  https://www.mapleprimes.com/posts/202437-Integer-Points-In-Polyhedral-Regions  will be useful for solving your problems.

Nested seq...

Kitonum 21475
September 21 2020
0 3

You can use nested  seq  for this. What we get is called not permutations but Cartesian product  


 

> 

restart;
S:={0,1,2}:
seq(seq(seq(seq([u,x,y,z], z=S), y=S), x=S), u=S);

[0, 0, 0, 0], [0, 0, 0, 1], [0, 0, 0, 2], [0, 0, 1, 0], [0, 0, 1, 1], [0, 0, 1, 2], [0, 0, 2, 0], [0, 0, 2, 1], [0, 0, 2, 2], [0, 1, 0, 0], [0, 1, 0, 1], [0, 1, 0, 2], [0, 1, 1, 0], [0, 1, 1, 1], [0, 1, 1, 2], [0, 1, 2, 0], [0, 1, 2, 1], [0, 1, 2, 2], [0, 2, 0, 0], [0, 2, 0, 1], [0, 2, 0, 2], [0, 2, 1, 0], [0, 2, 1, 1], [0, 2, 1, 2], [0, 2, 2, 0], [0, 2, 2, 1], [0, 2, 2, 2], [1, 0, 0, 0], [1, 0, 0, 1], [1, 0, 0, 2], [1, 0, 1, 0], [1, 0, 1, 1], [1, 0, 1, 2], [1, 0, 2, 0], [1, 0, 2, 1], [1, 0, 2, 2], [1, 1, 0, 0], [1, 1, 0, 1], [1, 1, 0, 2], [1, 1, 1, 0], [1, 1, 1, 1], [1, 1, 1, 2], [1, 1, 2, 0], [1, 1, 2, 1], [1, 1, 2, 2], [1, 2, 0, 0], [1, 2, 0, 1], [1, 2, 0, 2], [1, 2, 1, 0], [1, 2, 1, 1], [1, 2, 1, 2], [1, 2, 2, 0], [1, 2, 2, 1], [1, 2, 2, 2], [2, 0, 0, 0], [2, 0, 0, 1], [2, 0, 0, 2], [2, 0, 1, 0], [2, 0, 1, 1], [2, 0, 1, 2], [2, 0, 2, 0], [2, 0, 2, 1], [2, 0, 2, 2], [2, 1, 0, 0], [2, 1, 0, 1], [2, 1, 0, 2], [2, 1, 1, 0], [2, 1, 1, 1], [2, 1, 1, 2], [2, 1, 2, 0], [2, 1, 2, 1], [2, 1, 2, 2], [2, 2, 0, 0], [2, 2, 0, 1], [2, 2, 0, 2], [2, 2, 1, 0], [2, 2, 1, 1], [2, 2, 1, 2], [2, 2, 2, 0], [2, 2, 2, 1], [2, 2, 2, 2]

 (1)
> 

nops([%])=ifactor(nops([%]));

81 = ``(3)^4

 (2)
> 

 


 

Download nested_seq.mw

Example...

Kitonum 21475
September 19 2020
0 2


 

> 

restart;
opt1:="a=1, b=2":
opt2:="a=0.7, b=3":
f:=a*x^b:
plot([eval(f,[parse(opt1)]),eval(f,[parse(opt2)])], x=-2..2, color=[red,blue], legend=[opt1,opt2], legendstyle=[location=left, font=[times,16]], scaling=constrained, size=[600,600]);

 
> 

 


Edit: Here's another (prettier but more complex) version:

restart;
opt1:="a=1, b=2":
opt2:="a=0.7, b=3":
f:=a*x^b:
A:=plot([eval(f,[parse(opt1)]),eval(f,[parse(opt2)])], x=-2..2, color=[red,blue], thickness=2):
B:=plot([5.2,4.7], x=-0.9..-0.4, color=[red,blue], thickness=3):
T:=plots:-textplot([[-2.9,5.2,opt1],[-3.2,4.7,opt2]], font=[times,16], align=right):
plots:-display(A,B,T, scaling=constrained, size=[600,500]);

                          

 

Download legend.mw

indets...

Kitonum 21475
September 19 2020
2 0 
restart;
A:=Matrix([[0,1,-1],[-1,-2,2],[-1,-2,2]]):
v:=Vector([-1,1,1]):
sol := LinearAlgebra:-LinearSolve(A,v);
indets(sol, 'indexed');

 

op...

Kitonum 21475
September 19 2020
2 0 
v:=[2,4,7];
ilcm( op(v));
# Or
ilcm(v[]);

 

Design...

Kitonum 21475
September 18 2020
0 0

In the help for  fsolve  we see  
"variables - (optional) name or set(name); unknowns for which to solve"

So use  x0  and  y0  instead (for example).

Inert operation...

Kitonum 21475
September 17 2020
0 0


 

> 

restart;
> 

x%*x;

`%*`(x, x)

 (1)
> 

value(%);

x^2

 (2)
> 

 


 

Download inert.mw

paraboloid...

Kitonum 21475
September 15 2020
2 0
> 

restart;         
plot3d(eval([x,y,4*x^2+9*y^2],[x=1/2*r*cos(t),y=1/3*r*sin(t)]), r=0..1, t=0..2*Pi, axes=normal, scaling=constrained);

> 

 

Download _paraboloid.mw

Adjustment...

Kitonum 21475
September 10 2020
0 0

Use  LinearAlgebra:-GenerateMatrix  instead of  linalg:-genmatrix :

 

Download sys.mw

Adjustment...

Kitonum 21475
September 08 2020
0 1

In Maple the imaginary unit should be coded as  I  not  i :


 

> 

restart;
with(LinearAlgebra):
with(plots, implicitplot):
i:=I:
M := Matrix([[sigma + Gamma*i*k + 0.5 + 0.5*tanh(c - v) + beta*v, -0.5 + 0.5*tanh(c - v), 0.5*(w - u)*sech^2*(c - v) - beta*u], [-0.5 - 0.5*tanh(c - v), sigma - i*k*Gamma + 0.5 + 0.5*tanh(c - v) + beta*v, 0.5*(u - w)*sech^2*(c - v) - beta*w], [-beta*v, -beta*v, sigma - beta*alpha*(u + w)]]);

Matrix(3, 3, {(1, 1) = sigma+I*GAMMA*k+.5+.5*tanh(c-v)+beta*v, (1, 2) = -.5+.5*tanh(c-v), (1, 3) = (.5*w-.5*u)*sech^2*(c-v)-beta*u, (2, 1) = -.5-.5*tanh(c-v), (2, 2) = sigma-I*GAMMA*k+.5+.5*tanh(c-v)+beta*v, (2, 3) = (.5*u-.5*w)*sech^2*(c-v)-beta*w, (3, 1) = -beta*v, (3, 2) = -beta*v, (3, 3) = sigma-beta*alpha*(u+w)})

 (1)
> 

Eq := Determinant(M);

.50*tanh(c-v)*sigma+1.0*tanh(c-v)*sigma^2+.50*tanh(c-v)^2*sigma+1.0*sigma^2+sigma^3-1.0*sigma*tanh(c-v)*beta*alpha*u-1.0*sigma*tanh(c-v)*beta*alpha*w-1.0*tanh(c-v)*beta^2*v*alpha*u-1.0*tanh(c-v)*beta^2*v*alpha*w+.50*tanh(c-v)*beta*v^2*sech^2*u-.50*tanh(c-v)*beta*v^2*sech^2*w-Gamma^2*k^2*beta*alpha*u-Gamma^2*k^2*beta*alpha*w-2*sigma*beta^2*v*alpha*u-2*sigma*beta^2*v*alpha*w+Gamma^2*k^2*sigma+beta^2*v^2*sigma+2*beta*v*sigma^2+1.0*beta*v*sigma-1.0*beta^2*v*w-beta^3*v^2*u-1.0*beta^2*v*u-beta^3*v^2*w-(1.0*I)*Gamma*k*beta*v*c*sech^2*w+(1.0*I)*Gamma*k*beta*v*c*sech^2*u-beta^2*v*u*sigma-.50*tanh(c-v)*beta*alpha*u-.50*tanh(c-v)*beta*alpha*w-.50*tanh(c-v)^2*beta*alpha*u-.50*tanh(c-v)^2*beta*alpha*w+1.0*tanh(c-v)*beta*v*sigma-1.0*tanh(c-v)*beta^2*v*u-1.0*sigma*beta*alpha*u-1.0*sigma*beta*alpha*w-1.0*beta^2*v*alpha*u-1.0*beta^2*v*alpha*w-sigma^2*beta*alpha*u-sigma^2*beta*alpha*w-sigma*beta^2*v*w-beta^3*v^2*alpha*u-beta^3*v^2*alpha*w-(1.0*I)*Gamma*k*beta*v^2*sech^2*u+(1.0*I)*Gamma*k*beta*v^2*sech^2*w-.50*tanh(c-v)*beta*v*c*sech^2*u+.50*tanh(c-v)*beta*v*c*sech^2*w+I*beta^2*v*u*Gamma*k-I*Gamma*k*beta^2*v*w

 (2)
> 

alpha := 0.1;
beta := 0.01;
mu := 0.5;
u := 0.5;
v := 1;
Gamma := 0.1;
c := 2;
w := 0.5;

.1

  

0.1e-1

  

.5

  

.5

  

1

  

.1

  

2

  

.5

 (3)
> 

p1i := evalc(Re(Eq));

(-0.11110e-3+0.*I)+0.8980e-2*sigma-0.5600e-3*tanh(1)-0.500e-3*tanh(1)^2+.50900*tanh(1)*sigma+1.0*tanh(1)*sigma^2+.50*tanh(1)^2*sigma+0.1e-1*k^2*sigma+1.0190*sigma^2-0.10e-4*k^2+sigma^3

 (4)
> 

p1 := implicitplot(Eq, k = 0 .. 10, sigma = -0.01 .. 0.01);

 
> 

 


Edit.  evalc(Im(Eq))  returns  0. , that is  Eq  is real for real  k  and  sigma  and specific values the other parameters.
 

Download plot_new.mw

Another way...

Kitonum 21475
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 21475
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:

     

 

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