## 15998 Reputation

12 years, 188 days

## numpoints...

You have taken too little value for numpoints  (n=100). Try

`plot3d( [V,0], 0..2*Pi, 0..2*Pi, numpoints=40000);`

## Sum...

Use the inert sum, that is  Sum  instead of  sum :

```restart;
X := x->Sum((-1)^m*((1/2)*x)^(2*m+n)/(factorial(m)*factorial(m+n)), m = 0 .. infinity);
diff(X(x), x);  # The derivatives series
applyop((combine@expand),1,%,2*m+n); # Simplification
```

Edit.

## @@...

```f:=x->sqrt(1+x);
(f@@6)(x);```

## Procedure for this...

```restart;

Pbc:=proc(n::posint)
local Ind, Var1, Var2;
uses combinat;
Ind:=permute([0\$n,1\$n],n):
Var1:=[seq(x||i,i=0..n-1)]: Var2:=[seq(y||i,i=0..n-1)]:
{seq(seq(inner(Var1,ind1)*inner(Var2,ind2), ind1=Ind), ind2=Ind)};
end proc:
```

Examples of use (for n = 2 the result coincides with Carl's one):
Pbc(1);
Pbc(2);
Pbc(3);

## General solution...

There will be infinitely many such vectors. All of them belong to a plane perpendicular to the original vector. Therefore, the set of all solutions depends on two parameters:

```restart;
PV:=(a,b,c)->solve(a*x+b*y+c*z=0,{x,y,z}):

# Examples of use:
V:=<1,2,3>;
PV(V[1],V[2],V[3]);
V1:=eval(<x,y,z>, %); # General solution
V.V1; # Check
V2:=eval(V1,[y=1,z=1]);  # One of the solutions
```

## functions as procedures...

Define functions as procedures, not expressions:

```restart;
h:=x->1/(1+exp(-x));
hh:=(w,x,b)->ln(h(w*x + b));
diff(hh(w,x,b), w);  # Or  D[1](hh)(w,x,b);
normal(expand(%));
```

I always refer to previous objects using their names or by  %  and so on.

Edit.

## IntegrationTools:-Change...

Here is a more traditional way:

```restart;
IntegrationTools:-Change(int(arccos(x)*arcsin(x),x), x=sin(t));
eval(%, sin(t)=x);```

```restart;
b:=proc(n)
option remember;
if n=0 then return 0 elif n=1 then return 1
else
(b(n-1)+b(n-2))/2:
fi;
end proc:

# Examples of use:

S:=seq(b(n),n=0..10);
plot([\$ 0..10],[S]);```

You can obtain an explicit formula for the nth member by  rsolve  command:

```restart;
# The formula for nth term
rsolve({b(n)=(b(n-1)+b(n-2))/2,b(0)=0,b(1)=1}, b(n));
```

## bug...

This is certainly a bug. Do

```with(geometry):
point(o, 0, 0);
point(A, 0, 1);
point(d, 0, 2);
point(F, 0.8944271920, 1.4472135960);
line(lOD, [o, d]);
line(lAF, [A, F]);
alpha1 := FindAngle(lOD, lAF);
alpha:=min(alpha1,Pi-alpha1);```

alpha := 1.107148718

## @@ operator...

```f:=y->sqrt((1+y)/2):
cos(Pi/2)=0:
cos(Pi/2^n)=(f@@(n-1))(0):

# Examples
cos(Pi/16)=(f@@3)(0);
cos(Pi/64)=(f@@5)(0);
```

## Workaround...

Any command, if you add the  %  symbol in front of it, becomes inert. If we replace  Sum  with  %sum , then the bug disappears. See below

## Solution for specific parameters...

Your system contains too many parameters  Q11A, Q11Ay, Q11Az, Q11Iy, Q11Iz, Q11J, Q16A, Q16Ay, Q16Az, Q16Iz, Q16J, Q55A, Q55Ay, Q55Iy, Q66A, Q66Az, Q66Iz  (total 17 ones) with unknown values. If you specify the values of these parameters, Maple easily solves this system. I took parameter values from 1 to 17.

## Procedure...

Here is a procedure for this:

```LeastDegree:=proc(P::polynom, var::{set,list})
local L;
L:=sort(map(p->degree(p,var),[op(P)]));
select(p->degree(p,var)=L[1],P);
end proc:
```

Examples of use:

f:= 100*x^2*y^2 + 35*x^2*y + 45*y:
g:= 13*x^2*y^2 + x*a*y^2 + 2*y*x^2:
LeastDegree(f, [x,y]);
LeastDegree(g, [x,y]);

45*y
a*x*y^2+2*x^2*y

Edit.

## The range for x...

Specify the range for x:

`plot(sin(4*x)+1/3*cos(6*x), x=0..2*Pi);`

## Another animation...

This animation shows thin cylindrical disks, summing up the volumes of which and passing to the limit, we get the exact volume:

```restart;
f:=x^(1/2):
g:=x^2/8:
X:=r*cos(phi): Y:=r*sin(phi):
P:=plot3d(eval([[X,Y,f],[X,Y,g]],x=r), r=0..4, phi=0..2*Pi, style=surface, color=["Khaki","LightBlue"], scaling=constrained, axes=normal, labels=[z,x,y], orientation=[20,80], transparency=0.5):
F:=y->plots:-display(plot3d([[r*cos(phi),r*sin(phi),y],[r*cos(phi),r*sin(phi),y+h]], r=y^2..sqrt(8*y), phi=0..2*Pi, style=surface, color=gold), plot3d([[y^2*cos(phi),y^2*sin(phi),H],[sqrt(8*y)*cos(phi),sqrt(8*y)*sin(phi),H]], H=y..y+h, phi=0..2*Pi, style=surface, color=gold)):
h:=0.15:
plots:-animate(F,[y], y=0..2-h, frames=60, background=P);
```

# Volume of this body in 2 ways
Int(Pi*(8*y-y^4), y=0..2)=int(Pi*(8*y-y^4), y=0..2);  # Washer (disk) method
Int((sqrt(x)-x^2/8)*2*Pi*x, x=0..4)=int((sqrt(x)-x^2/8)*2*Pi*x, x=0..4);  # Shell (cylinder) method

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