@Carl Love 

Thanks

That is is nice thing of calculus the geometric/graphical way for seeing the big picture. 

Remember me solving long ago by hand with the help of the "hessian"  determinant  to investigate  the special points of a f(x,y) function, so some basics i do know. 

Three conditions :A ,B and C must fulfilled to know what type of special point is concerned for f(x,y)
For a extreme point (condition B)  you need the hessian determinant.  

The hessiaan determinant is filled with : with second order partial deratives and two in the determinant are equal ..and so on .
Got here old studymaterial what makes you think  : 
For example : Determine the conditions for a function with a maximum in (a1,a2)
Indeed i do see also now the gradient=0  as conditions here, but did not memorize it. 

Geometrical interpretation ..i try this with the complex numbers and their functions : differentating and integrating
Is there a serie of lessons to find for  Maple for complex numbers and analyse?

GettingStarted.pdf

Added also the Getting Started helpfile frorm where it must be possible according to the developer to install the calculus package.
Well, id not get it clear from these instructions what exactly to install for Maple 2021 ?

@janhardo 

I could add the partial deratives from  f(x,y)  in x and y direction, directional derative and gradient also in the 3D plot 
Showing also the level curve and the gradient as vectorfield from  f(x,y)
A while ago i do want to know more about vectorfields and ended at a tensor from vectorcalculus (if i am correct) , that becomes too complex 

@Carl Love 

The gradient vector is in a stationary point of a function of two variables the value 0 ( no vector : the zero vector) 
The stationary points thus found through the gradient are examined with the second partial derivative in those stationary points ( this is the "second derivative test" )

Now I know why the gradient is defined in the procedure 

@janhardo 

I like to see also a procedure made out of the calculation made by @Kitonum and @Acer if possible?

Better is trying to make it by meself first.

@Carl Love 
Thanks
Unfortunaly, no idea how this procedure is constructed 
See some sub procedures contained in a main procedure, but that's all 
How the sub  procedures are constructed ?
How the main procdure is constructed with the automatic scaling ?

Its complicated, because i am not really strong in vector calculus , so defining for example a gradient in terms of partial deratives ..?

Gradient:= proc(f)
local x, y, V:= (x,y); 
    unapply(diff~(f(V), [V]), [V])
end proc
:

For a 3D plot it is a levelcurve and standing on this is the gradient vector , and don't know why is used a gradientvector in the procedure

@acer 

We do now know the steps to solve this.

I think in general if you want to automate calculations with a procedure out of these calculations : what are the considerations ?
What are the steps to take.  

@acer 

The green points are the saddle points and drawing two levelcurves there shows exactly what characterise these saddle points

@acer 

Thanks

For a procedure to make there is the function itself and the domain to specify as input 
So a message : enter function and domain.
For educational purpose : showing the steps in the calculation as a list 
Showing in the critical points the two tangentlines and  levelcurves

Also interesting is to go  further with a plot of two functions also in a procedure and do something with a gradient and level curves

@Kitonum 

Thanks

Its good to know in your worksheet how those critical points are named.
And there is are locals points  and  more.

@Rouben Rostamian  
Thanks
There are for sure 6 points, but we don't nothing yet about the sadlepoints.
Perhaps later adding some more plots in one view to get the whole picture of

For now i look in old studymaterial how the questions were formulated.

@nm 
Thanks

Looks good to read!
The axis need also a symbol 
Comes later..

Taking_a_list_of_numeric_values_L.mw

@acer 
Thanks

[[0, f(0)], [Pi/6, f(Pi/6)], [Pi/4, f(Pi/4)], [Pi/3, f(Pi/3)], [Pi/2, f(Pi/2)]]

this is a list of lists of sin(x), without the functionvalues, but when i do want this list? 

@Kitonum 

Thanks, its clear