@janhardo 
Unfortanely could not follow complete the handling of the partial deratives  and integrating

In general integrating in 1,2 and 3 variables and their constants?
What says that one partial derative is zero in u=f(x,y,z) ?

Could this figure out in Maple symbolically ?..yes it can, but how
With pdsolve( two or more variables) and dsolve (one variable)

example

A := exp(x)*cos(y) + y*z

f := int(A, x) + g(y, z);
              f := y z x + exp(x) cos(y) + g(y, z)

The constant is here g(y,z) 

Now diff(f,y) should be B. Obtain that derivative and compare to B.

diff(y*z*x + exp(x)*cos(y) + g(y, z), y) - B = 0;
                         d             
                        --- g(y, z) = 0
                         dy            

This says g(y,z) is independent of y, so g(y,z) = h(z), a function of z only. That makes f become

ff := eval(f, g(y, z) = h(z));
               ff := y z x + exp(x) cos(y) + h(z)

@rlopez 

Thanks

Well, if you have reached all goals in life, then this person is a satisfied (enlighted) person i belief.
( maslow pyramid)

I noticed for your worksheet that it is maple 1d input in a 2D document and this is not how a regular student tackle his math
i think.
But i like the direct style, but for that is needed  a some more working knowledge of Maple commands     
 

@rlopez 
Thanks!

Interesting see another solving method and helpful.
I figured out how to do it in 2D mode too 
Only try to follow a bookexample and there is no curl  there mentioned
Picked up :  it is conservative field and those filelds are not rotating, so curl = 0 then    

That's a long time ago 50 years back, at that time i was a teenager of 16 years old

thomas_calculus-vb2-blz_1075_-vraag_forum_deel2.mw
   

@tomleslie 

Thanks!

With pdsolve you get the complete general function and with ScalarPotential you must add constant to the equation
Did not studied yet the outcome of a PDE to a serie of ODE 
But the idea is to this do it by first principles in 2D mode, like a student must do 
 

With the documentmode is it possible to get quick idea of a pattern, but beware.
For example i integrated  for f(x,y,z)= x+y+z with respect to x 
The pattern is that x ( whatever function x is ) is integrated and  y and  z are multiplied with only x  
x + y + z  -> int = 1/2*x^2 + y*x + z*x  
2*x + y + z-> int =x^2 + x*y + x*z
x*sin(x) + y + z -> int= y*x + z*x + sin(x) - x*cos(x)

That's brings the question can i see in documentmode to respect what variable is diferentiate/integrate , because there is no notation of it

You can do this for integration/differentation for the 3 variables x,y,z to get a idea how the final function is derived from the vectorfunction 

Or the idea is 
F is vectorfunction (given) => gradF is gradientfield  = dF/dx, dF/dy,dF/dz (round d) 
So integrate for the 3 partial deratives gives a scalar function back ( no need to look for pattern  for integrate/differentate)  

@9009134 

If you click on plot ( in document mode) , you get a context menu. There is a tickbox  with scaling constrained option too.
You can experiment with other plot options there. 

@vv 
Thanks 
For the good advice.
Understanding a math concept is the first step and then practicing by hand 
To do it on your own makes it even harder and probably for many math topics impossible without a teacher
But these day internet can be source of  studymaterial. 
Must be possible for me think as B Ed in math to study some calculus topics, if am taking the time for it of course.
 

@vv 

It is a sparringpartner Maple for understanding more  calculus, but it follows not always a textbook for integration for not adding a constant
 

@vv 

Thanks

There are two functions involved f(x) and f(y) for  z= f(x,y)  for partiel differentation
(If you consider it as two intersections curves from a surface as defining a partiele directive)
So then f[x](x,y) =0    ..y(x) is left over as constant

It s getting more complicated in another example with v(x,y,z) for a vectorfield:.has it a potential function?

@tomleslie 

Thanks

Its solved as you mentioned : there is a display command needed 

I execute a animate command without display in the table 

All is working fine i think 

@tomleslie 

Thanks

Indeed

The first worksheet attached if you take this one or take the one  from dr Lopes the output of a intermediate animate plot gives this message.

But the final animate works.   

@tomleslie 

What should i could  aspect  for the arttached worksheet ?
Most all of the commands are suppresed by a colon and focus is on the animation of the positionvector   

When i do a iniline with Ctrl+=  for P1 there is message : 

          [Length of output exceeds limit of 1000000]

So this animation cannot be seen , but the final anmation works 

@tomleslie 

Thanks

Your fix for method 3 shows a spatial curve for the positionvector, while i thought i was for 2D ?  

@rlopez Y

Thanks

Did not know that there was non-executable math applied for Method 3 for the worksheet
Well it is now up to date the worksheet and what direction could Maple2020 go, that this revised worksheet for 2020 will not be working again for the future ?

Must be a interesting list of math topics. 

@janhardo 

I bought calculus III lessons

Its very useful an it covers a lot ,although the vectorcalculus packgage is not used here because 
its old lessonmaterial. 
You can study some vectorcalculus this lessons   

@tomleslie 
Thanks!
None of the examples are diificult to understand in both modes,
They are the same ,but in 1D mode it takes some more work for programmer input