As I understand it, the OP's meaning is exactly that: using two independent variables called z and z bar. The problem is that for Maple zbar is represented by the function call conjugate(z). So, it is a system design issue, not a mathematician vs physicist one.

As I understand it, the OP's meaning is exactly that: using two independent variables called z and z bar. The problem is that for Maple zbar is represented by the function call conjugate(z). So, it is a system design issue, not a mathematician vs physicist one.

Actually I would say how d |z|/dz can be computed by hand (for what  I use latexlike notation). Whether computationally it would be useful as a definition extending the current abs(1,z) to z complex may be another issue.

On one hand, d|z|^2/dz = 2 |z| d |z|/dz. On the other hand, as |z|^2 = z \bar{z}, d|z|^2/dz = \bar{z}, as d \bar{z} / dz =0 because z and \bar{z} are independent variables. So, d |z|/dz =  \bar{z} /(2 |z| ) (note that the d's here should be read as \partial ).

Now z and \bar{z} are independent in analogous sense that x and y, the real and imaginary parts of z are independent. E.g. you can write relate the differentials by the equations dz = dx + i dy and d \bar{z} = dx - i dy, hence solve for dx and dy. This is not the same as the pair z and -z, because d (-z) = - dx - i dy, not independent from dz.

Actually I would say how d |z|/dz can be computed by hand (for what  I use latexlike notation). Whether computationally it would be useful as a definition extending the current abs(1,z) to z complex may be another issue.

On one hand, d|z|^2/dz = 2 |z| d |z|/dz. On the other hand, as |z|^2 = z \bar{z}, d|z|^2/dz = \bar{z}, as d \bar{z} / dz =0 because z and \bar{z} are independent variables. So, d |z|/dz =  \bar{z} /(2 |z| ) (note that the d's here should be read as \partial ).

Now z and \bar{z} are independent in analogous sense that x and y, the real and imaginary parts of z are independent. E.g. you can write relate the differentials by the equations dz = dx + i dy and d \bar{z} = dx - i dy, hence solve for dx and dy. This is not the same as the pair z and -z, because d (-z) = - dx - i dy, not independent from dz.

In fact it is "partially" defined. For z real, where conjugate(z)=z (so that they are not independent variables), it is right:

diff(conjugate(z),z) assuming real;
                                  1

Otherwise, so that the implicit assumption that z is complex holds, hence conjugate(z) is independent from z, the output:

diff(conjugate(z),z);
f:=unapply(%,z):
                            _
                            z    2 abs(1, z)
                         - --- + -----------
                            z     signum(z)

is not a priory wrong, as this expression is actually 0 if simplified by hand, but Maple does not make this simplification, in particular as abs(1,z) is not defined for nonreal z:

f(I);

Error, (in simpl/abs) abs is not differentiable at non-real arguments

In fact it is "partially" defined. For z real, where conjugate(z)=z (so that they are not independent variables), it is right:

diff(conjugate(z),z) assuming real;
                                  1

Otherwise, so that the implicit assumption that z is complex holds, hence conjugate(z) is independent from z, the output:

diff(conjugate(z),z);
f:=unapply(%,z):
                            _
                            z    2 abs(1, z)
                         - --- + -----------
                            z     signum(z)

is not a priory wrong, as this expression is actually 0 if simplified by hand, but Maple does not make this simplification, in particular as abs(1,z) is not defined for nonreal z:

f(I);

Error, (in simpl/abs) abs is not differentiable at non-real arguments

I do not think so. All the books on complex variable that I am aware of, and that were written by mathematicians, do use differentiation wrt \bar{z}, which in Maple is represented by the function call conjugate(z).

I do not think so. All the books on complex variable that I am aware of, and that were written by mathematicians, do use differentiation wrt \bar{z}, which in Maple is represented by the function call conjugate(z).

What I can get out of your messy post is that you have made subtitutions like:

subs((D[1](f))(x, y, z) = f[x], ee); 

replacing function calls, like D[1](f)(x, y, z), for indexed names like f[x]. The former contains information on the dependence, while the latter does not.

What I can get out of your messy post is that you have made subtitutions like:

subs((D[1](f))(x, y, z) = f[x], ee); 

replacing function calls, like D[1](f)(x, y, z), for indexed names like f[x]. The former contains information on the dependence, while the latter does not.

One way:

restart:
u :=  (x, y, z)-> (diff(f(x, y, z), x))*(diff(f(x, y, z), y)):
diff(u(x, y, z),z):
subs(diff(f(x,y,z),x,z)=diff(phi(x, y, z, fx, fy),x),
diff(f(x,y,z),y,z)=diff(phi(x, y, z, fx, fy),y),%);

  /d                      \ /d            \
  |-- phi(x, y, z, fx, fy)| |-- f(x, y, z)|
  \dx                     / \dy           /

           /d            \ /d                      \
         + |-- f(x, y, z)| |-- phi(x, y, z, fx, fy)|
           \dx           / \dy                     /

One way:

restart:
u :=  (x, y, z)-> (diff(f(x, y, z), x))*(diff(f(x, y, z), y)):
diff(u(x, y, z),z):
subs(diff(f(x,y,z),x,z)=diff(phi(x, y, z, fx, fy),x),
diff(f(x,y,z),y,z)=diff(phi(x, y, z, fx, fy),y),%);

  /d                      \ /d            \
  |-- phi(x, y, z, fx, fy)| |-- f(x, y, z)|
  \dx                     / \dy           /

           /d            \ /d                      \
         + |-- f(x, y, z)| |-- phi(x, y, z, fx, fy)|
           \dx           / \dy                     /

In extended typesetting, a set or list of the Matrix symbols also remain unevaluated, but somewhere else the evaluation occurs:

convert(indets(gradV),list);
ToInert(%);
                        [a11, a12, a21, a22, x1, x2]

_Inert_LIST(_Inert_EXPSEQ(_Inert_INTPOS(1), _Inert_INTPOS(0), 
  _Inert_INTPOS(0), _Inert_INTPOS(1), _Inert_NAME("x1"), 
_Inert_NAME("x2")))

My interpretation is that the evaluation occurs in the kernel as usual, but the GUI runs in parallel a kind of second kernel where evaluation does not occur.

A probably related issue with evaluation under extended typesetting was described in this thread.

In extended typesetting, a set or list of the Matrix symbols also remain unevaluated, but somewhere else the evaluation occurs:

convert(indets(gradV),list);
ToInert(%);
                        [a11, a12, a21, a22, x1, x2]

_Inert_LIST(_Inert_EXPSEQ(_Inert_INTPOS(1), _Inert_INTPOS(0), 
  _Inert_INTPOS(0), _Inert_INTPOS(1), _Inert_NAME("x1"), 
_Inert_NAME("x2")))

My interpretation is that the evaluation occurs in the kernel as usual, but the GUI runs in parallel a kind of second kernel where evaluation does not occur.

A probably related issue with evaluation under extended typesetting was described in this thread.

Simpler:

dsolve(de,implicit):
collect(%, x);
                   3
                  x           2
               - ---- + (-y(x)  + 2) x + _C1 + y(x) = 0
                  3