Difficulties in the operator notation in partial differentiation While trying partial differentiation, I came to a dead end, where the book didn't provide me a satisfactory explanation. First of all, what does the notation mean: Say I have a relation
$$x^2{\partial z\over \partial x}+y^3{\partial z\over \partial y}={\partial^2 z\over \partial t^2}$$
Then the partial notation yields me with
$$x^2{\partial \over \partial x}+y^3{\partial \over \partial y}={\partial^2 \over \partial t^2}$$
I don't know whether this is documented or not, but often writers use this stuff. My main query lies in the evaluation of parentheses having these operators. Some writers use
$$(x{\partial \over \partial x}+y{\partial \over \partial y})^2z=x^2{\partial^2 z\over \partial x^2}+xy{\partial^2 z\over \partial x\partial y}+xy{\partial^2 z\over \partial y\partial x}+y^2{\partial^2 z\over \partial y^2}$$
But I feel doing it this way
$$(x{\partial \over \partial x}+y{\partial  \over \partial y})(x{\partial \over \partial x}+y{\partial \over \partial y})z=(x{\partial \over \partial x}+y{\partial  \over \partial y})(x{\partial z\over \partial x}+y{\partial z\over \partial y})$$
Since there is an operator before, I feel like separating the operators and this gives me
$$(x{\partial \over \partial x}+y{\partial  \over \partial y})(x{\partial z\over \partial x}+y{\partial z\over \partial y})=x{\partial \over \partial x}(x{\partial z\over \partial x}+y{\partial z\over \partial y})+y{\partial \over \partial y}(x{\partial z\over \partial x}+y{\partial z\over \partial y})$$
This however gives me
$$(x{\partial \over \partial x}+y{\partial \over \partial y})^2z=x^2{\partial^2 z\over \partial x^2}+xy{\partial^2 z\over \partial x\partial y}+xy{\partial^2 z\over \partial y\partial x}+y^2{\partial^2 z\over \partial y^2}+x{\partial z \over \partial x}+y{\partial z \over \partial y}$$
What's going wrong in here?
 A: The notation that you are using appears inconsistent to me. For example you write
$\displaystyle(x{\partial \over \partial x}+y{\partial \over \partial y})^2z=x^2{\partial^2 z\over \partial x^2}+xy{\partial^2 z\over \partial x\partial y}+xy{\partial^2 z\over \partial y\partial x}+y^2{\partial^2 z\over \partial y^2} \tag{1}$
which I disagree with. I could accept the interpretation of the left hand side of (1) as either applying the operator twice to $z$ in succession, or it could be the result of squaring the application of the operator to $z$, e.g.
$A(z) = (\displaystyle x{\partial z \over \partial x}+y{\partial z \over \partial y})$
then
$\displaystyle(x{\partial \over \partial x}+y{\partial \over \partial y})^2z$
could either mean
a) $A( A(z) )$
or
b) $A(z)^{2}$.
I would be more comfortable with it meaning a), as this is more consistent with the use of the squared sign, and also
$A( A(z) ) =\displaystyle x^2{\partial^2 z\over \partial x^2}+xy{\partial^2 z\over \partial x\partial y}+xy{\partial^2 z\over \partial y\partial x}+y^2{\partial^2 z\over \partial y^2}+x{\partial z \over \partial x}+y{\partial z \over \partial y}$
However, the right hand side of (1) cannot be obtained using either a) or b), and I would suggest not to use this notation when writing documents
