## 1400 Reputation

18 years, 300 days
University of Twente (retired)
Enschede, Netherlands

My "website" consists of a Maple Manual in Dutch

## Allsolutions...

The exponential function a^b is defined as exp(b*ln(a)). If a<0, than ln(a) is the multi-valued complex function ln(-a)+πi+2kπi. You can find these by using the option allsolutions:

 > restart;
 > s := solve(diff(-1/x,x) = (-1/x)^(b), b, allsolutions);
 (1)
 > simplify(eval(s,_Z1=1),symbolic);
 (2)
 >

For the second equation there seems do be no solution independent of x.

## Elliptic domain...

You may define the vectorfield [0,0] outside the ellipse:

 > restart; with(plots):
 > f := (x,y) -> if x^2+y^2/2 <= 1 then x^2+y^2 else 0 end if:
 > g := (x,y) -> if x^2+y^2/2 <= 1 then x+y else 0 end if:
 > fieldplot( [f,g], -1.5..1.5,-1.5..1.5, arrows=slim );
 >

## Perhaps you mean......

I suppose that you want to plot a parametrized surface, that is a list [x(u,v),y(u,v),z(u,v)].
This van be done ad follows:

 > restart;
 > S := (u,v) -> 10*u^2*v+20*u*v+15:
 > p:= proc(u,v) if u
 > plot3d([40*u,80*v,'p(u, v)'], u = 0 .. 1, v = 0 .. 2*u); #use quotes to prevent premature evaluation
 >

## Apply...

The inverse of unapply is apply. You have to provide a name for the independent variable:

 > a:=x -> x^2+sin(x);
 (1)
 > b := apply(a,t);
 (2)

or, simpler:

 > b := a(t);
 (3)

## This is not a Maple question...

First condition: F(exp(t)) = t, so F in the inverse of exp, that is F(t) = ln(t).

But ln(ln(exp(t))) = ln(t) ≠ 0

## Some suggestions...

(1) Make vectors A := <2,-3,a>: etc.

(2) A is perpedicular on B and C , so A.B=0 and A.C=0. These are two equations in in the unknown a,b,c.

(3) Make the third equation and solve.

## Initial conditions...

I suppose that you mean: y(t)=y'(t)=0 for all t<0. This can be implemented bij setting the initial condition at an arbitrary t<0:

 > restart;
 > eq := diff(y(t),t,t) + 9*y(t) = Dirac(t):
 > icon := y(-1)=0, D(y)(-1)=0:
 > dsolve( {eq,icon}, y(t) );
 (1)
 > dsolve( {eq,icon}, y(t), method=laplace );
 (2)
 >

The second solution that Maple gives uses:

 > int(Dirac(t),t=0..infinity );
 (1)
 >

## Maximize...

In the Optimization package you must use Maximize (capital M).

I suppose you want theta[1] and theta[2] in terms of p. Then there are only the trivial solutions (multiples of Pi):

evalc((M1.M2)(1,1)):  PP := evalc(Re(%)) assuming 0<p ,p<1:
solve({diff(PP,theta[1]),diff(PP,theta[2])},{theta[1],theta[2]},allsolutions);

{theta[1] = Pi _Z2, theta[2] = Pi _Z1}

## Another way...

Use an auxiliary variable:

```esp1 := -a^4+a^2*c^2: subs(q=a^2,factor(algsubs(a^2=q,esp1)));```

## Avoid zero polynomial...

You could make a procedure that rejects zero polynomials:

```f := proc(x::evaln)   local q:   q := randpoly(x, dense, degree = 1, coeffs = rand(-2 .. 2)):   if q=0 then f(x) else return q end if end proc;```

If you need a polynomial with only non-zero coefficients:

```f := proc(x::evaln)   local q:   q := randpoly(x, dense, degree = 1, coeffs = rand(-2 .. 2)):   if nops([coeffs(q)])=2 then return q else f(x) end if end proc;```

## Use expand or eval...

`subs((D(f)) = (t -> 1-t), Ex1): expand(%);`

I wonder why

`expand(subs((D(f)) = (t -> 1-t), Ex1));`

doesn't work.

Edited:

Or use:

`eval(Ex1,(D(f)) = (t -> 1-t));`

## Use of semicolons:...

if IT<20 then
IT:=IT+1;
if ZUM<(0.1)^(8) then IT=20
else
for i from 1 to 10 do YU[i]:=U[i] od;
for i from 2 to 9 do DU1:=YU[i+1]- YU[i-1] od:
end if;
ZUM:=U[1]-YU[1];
end if;

## Extract the 3rd solution...

If you want to go on and extract the 3rd solution (number only), you could simply do:

sol:=solve([sin(t), 0 < t, t < 8*Pi], t, allsolutions, explicit):
subs( sol[3], t );

## Pi or pi...

Use pi := 0.45;, because the constant π is represented as Pi (with capital P).

## ListTools,Search...

If you only need a sequence of position numbers:

L := [x,z,y]:
ListTools:-Search(y,L);
3
seq( ListTools:-Search(k,L), k=[x,y,z] );
1, 3, 2

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