## 18909 Reputation

14 years, 34 days

## Rotation the letter D in 2D...

@das1404  You can see here an example of creating and rotating in 2D the letter D. It seems you did not see this my answer. I'm wondering whether this file   LetterD1.mws  will work in Maple 7?

@waseem   Change the last line to

```plots:-display(Vector([seq(plot([seq(eval(lambda(F), Nb=j), j=[0.1,0.2,0.3])], delta2=0.02..0.1, legend=[Nb=0.1,Nb=0.2,Nb=0.3], title=('F'=F)), F=[10,20,30])]), size=[900,500]);
```

## Not necessary...

@ganelon  From this specification nothing will change: the solution is unique for any  y>0

## Impressive result...

@acer  How did you know about [Gr,Br] ?

## OK...

@Carl Love  You're right.

## Common options at the end of display com...

@das1404   If I need to draw several graphs on one plot, then I specify the common options (axes, scaling, view) only at the end of  plots[display]  command. In this case there will be no contradiction between the options.

We can simplify a little more by

```EXPR2:=combine(EXPR1, ln) assuming h>0, R0>0;
length(EXPR2);
```

## No problem...

@alfarunner  There are no problems here. The coefficients before rho hat and phi hat are the coordinates of the vector v(t) in this basis (here phi=t):

restart;
f:=unapply(<r(t)*cos(t),r(t)*sin(t)>, t):
v:=unapply(diff~(f(t),t), t):
V:=<cos(t),sin(t); -sin(t),cos(t)>.v(t);
simplify(V);
# The coordinates in the new basis (rho hat and phi hat)

I have never worked in the Physics package.

## Switch to the new coordinates...

@alfarunner  Above I wrote how to calculate the coordinates of the velocity vector in standard Cartesian coordinates. If you need the same, but in other coordinates, you just need to do a coordinate transformation, multiplying by the corresponding matrix:

restart;
f:=unapply(<r(t)*cos(t),r(t)*sin(t)>, t):
v:=unapply(diff~(f(t),t), t):
V:=<cos(t),sin(t); -sin(t),cos(t)>.v(t);

f:=unapply(<t*cos(t),t*sin(t)>, t):  # The same example
v:=unapply(diff~(f(t),t), t):
v(t);
v(1);
simplify(<cos(1),sin(1); -sin(1),cos(1)>.v(1));
plots:-display(plot([convert(f(t),list)[],t=0..3], color=red, thickness=3), plots:-arrow(f(1),v(1), width=[0.01,relative=false],head_width=[0.08,relative=false],color=blue), plots:-arrow(f(1),<cos(1),sin(1)>, width=[0.01,relative=false],head_width=[0.08,relative=false],color=black), plots:-arrow(f(1),<-sin(1),cos(1)>, width=[0.01,relative=false],head_width=[0.08,relative=false],color=black), scaling=constrained);

## Design...

@Preben Alsholm  Thank you for clarifying. Yet from the standpoint of pure mathematics it seems that this is not a very good design.

## Procedure...

@minhthien2016  To do this, the code should be rewritten as a procedure. After that, you can apply this procedure to any cube by specifying as a procedure argument the list of cube vertices.

Here is this procedure:

restart;
RegTetr:=proc(L)
local T, Dist, P;
uses combinat;
T:=combinat:-choose(L, 4);
Dist:=(p,q)->sqrt((p[1]-q[1])^2+(p[2]-q[2])^2+(p[3]-q[3])^2);
P:=v->combinat:-choose(v, 2);
[seq(`if`(nops(convert((Dist@op)~(P(t)),set))=1, t, NULL), t=T)];
end proc:

Example of use:

L:=[[0,0,0], [1,0,0], [1,1,0], [0,1,0], [0,0,1], [1,0,1], [1,1,1], [0,1,1]]:
L1:=2*~(map(t->t-~[1/2,1/2,1/2], L));
RegTetr(L1);

## A slightly different way...

@vv  Your solution is impressive, but weakly motivated (much remains behind the scenes). Why did you take these ranges  x=50..100, y=30..100, H=400..500 ? In addition, we can not so round up  y=40.00000000, because we obtain 0 under the logarithm sign.

Here is another solution in which we first eliminate  H , simplify the obtained system , and then decide, taking as the initial point x=41, y=41, because obviously should be  x>40, y>40  for real solution :

restart;
Digits:=1000:
Sys:=[21000 = ((10/1000)/60)*4181*983*(x-y),
H = -((10/1000)/60)*4181*ln(1-((x-y)/(x-40))),
H = 124+102*ln(x-40)+(7+2*ln(x-40))];
A:=eliminate(Sys, H):
Sys1:=convert(A[2], list):
Sys2:=expand(Sys1) assuming x>40, y>40;
fsolve(Sys2, {x=41,y=41}):
Sol:={%[], eval(Sys[3],%)};
# The final result
eval(Sys, Sol):  # Check
evalf[20](%);

## signum...

@Aaeru Michi  Maple 16 just does not cope with this example (I solved in Maple 2017). Here are 2 ways that maybe help you:

minimize(V, x = 0 .. 1, location) assuming x>0;  # or

minimize(-(1/2)*abs(x)*(c*x^2-2*L+x)/(2*c*x+1)^2, x = 0 .. 1, location);  # I just removed  signum(2*c*x+1)  from your example

## Optimization...

@Aaeru Michi   From help: "The Optimization package is a collection of commands for numerically solving optimization problems, which involve finding the minimum or maximum of an objective function possibly subject to constraints.  The package takes advantage of built-in library routines provided by the Numerical Algorithms Group (NAG)"

See help on  ?Optimization  for details.

## No...

Look at this:

A := plottools[polygon]([[sqrt(3)+1, 0, 0], [1, 1, 0], [1, -1, 0]], color = "Violet"):

B := plottools[polygon]([[sqrt(3)+1, 0, 0.01], [1, 1, 0.01], [1, -1, 0.01]], color = "Green"):

These are two different polygons at the distance = 0.01  one from the other.

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