@Ronan  In the MaplePrimes editor, click this button

@Ronan   See update to my answer.

@vv  Thank you for this example. It was quite unexpected for me. This is very instructive, because allows more critical attitude to  RootOf  representation for roots of equations.

@9009134  I do not know the answers to your questions.

@dumin 

f1:=8.044048-0.764286*x1-0.756746*x2+0.034524*x1*x1+0.022222*x2*x1+0.098413*x2*x2;
maximize(f1, x1=3..9, x2=1..7, location);

Addition.  If we consider  f1 on the whole plane  R^2, then the function is unbounded from above, i.e. can take any arbitrarily large values.

@Carl Love I think that the reason for the bug is related to the property of the function  Ei(x)  at  x=0. It would be better to add more  assuming epsilon>0  to my code.

Addition. I just noticed that the bug only occurs if the lower limit of the improper integral is 0. In other cases, the integral is calculated correctly, for example:

restart;
int(exp(-t)/(1-t), t = -0.1 .. infinity, CauchyPrincipalValue = true);
int(exp(-t)/(1-t), t = 0.1 .. infinity, CauchyPrincipalValue = true);
                                 0.7973339105
                                 0.5969990665

You forgot to call  plots  package in the last 3 lines of the code.

@vv  Thank you for the amendment. Of course the name of the procedure  FrechetDistance  is misleading. I changed it to  d .

@deniscr  In Maple 2018.1 it works properly. See file

dot_product.mw

In Maple 2017.3 it works also.

@digerdiga  

1. You specified formulas for the mapping r = sqrt (x * y) , t = ln (sqrt (x / y)), but did not specify on which set this mapping is defined. For example, it is clear that this set can not be a square  [0,1] x [0,1].

2. No.

3. Let's finish this rather meaningless conversation.

@digerdiga  You wrote "Are you sure? Because the mapping  r=sqrt(x*y), t=ln(sqrt(x/y))

seems fine to me..."

In the definition of a function  f: X -> Y , it is important not only the rule that each x of X is associated with some y of Y, but also the sets X and Y themselves. For the existence of an inverse function the correspondence between sets must be one-to-one ... Read carefully my previous comment.

@nm  See the first paragraph in help:

"discont returns a set of values where it is possible (not necessarily certain) that discontinuities occur."

Similarly:

discont(sqrt(1-x^2), x);
                                            {-1, 1}

@digerdiga 

1. Under the mapping  (x,y) -> (x*exp(y), x*exp(-y))  the straight lines  x=const<>0 are mapped onto hyperbolas and if  x=0  into one point  (0,0) . The straight lines  y=const  are mapped onto lines that pass through the origin.
Maple makes a bug when mapping a filled square.

2. There is no inverse mapping to the mapping  (x,y) -> (x*exp(y), x*exp(-y))  on the square  [0,1] x [0,1]   , because it is not one-to-one.

3. You wrote "Is there something like linspace(0,1,10) ? using seq([x, t, t = 0 .. 1], x = 0.1e-1 .. 1, 0.5e-1) does not have the other endpoint at 1, because the intervals do not match up to 1."  I did not understand the meaning of this phrase.

@digerdiga  1. Yes you are right. F(S) is enough.

2. Here is my first option:
restart;
S:=plottools:-rectangle([0,1],[1,0], color=red):
f:=(r,t)->[r*exp(t),r*exp(-t)];
F:=plottools:-transform(f):
plots:-display(F(S), scaling=constrained);
 

Maple incorrectly represents one side of a triangle as a straight line, it is easy to verify that it is not straight. 

3. When you write  x=r*exp(t), y=r*exp(-t), it means that  x  and  y  are just the coordinates of the point at which the point  (r,t)  maps under the indicated mapping. Instead  x  and  y , you can write for example  u  and  v  that does not change the point. In order to avoid misunderstandings, it is better to write so  (x,y) -> (x*exp(y), x*exp(-y))

@digerdiga  I first tried another option where it was necessary. Now everything is fixed.