Can I avoid using letters to denote partial derivatives of a function? Suppose $F$ is a two variable function, defined by $F(x,y)=x^2+y$. We can then calculate the partials of $F$ as $F_x(x,y)=2x$ and $F_y(x,y)=1$. Then it is clear that $F_x(1,0)=2$ and $F_y(0,1)=1$. That is to say that we can find expression for the partials of a function and then evaluate its values at different points by a simple substitution. 
If one wishes to find the derivative of $u^2+v$ with respect to $u$, it wouldn't make much sense to define an entirely new function involving $u$ and $v$ when we have a function whose expression at $x=u$ and $y=v$ looks exactly the same. So the derivative is $F_x(u,v)=2u$. So far I don't have any complaints. 
But what if instead we had the expression $y^2+x$? We should follow the same procedure and arrive at the correct result; but it just feels wrong to treat $x$ and $y$ as independent variables used to define a function at first, and then as merely a point at which we are evaluating a function (which is again given in terms of the independent variables $x$ and $y$). 
I feel that something like a list of independent variables used to define a function should have no form (such as denoting them by $x,y,z,$ etc). We need only know their expressions at an arbitrary point, say $(x,y)$. 
I also think that a function shouldn't exclusively be of some variables $x,y,z$ when there is no context. When "distance" is defined we may talk about velocity as a function of "distance". But when we want to define a function of arbitrary variables, why give them names? Why not say that $G$ is a 3 variable function and $G(x,y,z)=xy+z$? Also, this would mean that we shouldn't use letters to indicate the variables with respect to which partial derivatives have been taken. Any thoughts will be very much appreciated. Thank you.
 A: You are absolutely right: it is useful to be able to denote the partial derivatives of a function $f:\mathbb R^n\to\mathbb R$ without having to name its arguments, just as it is useful to be able to denote the derivative of a one-variable function $g:\mathbb R\to\mathbb R$ as $g':\mathbb R\to\mathbb R$. After all, $g'(0)$ is neither $\mathrm dg(0)/\mathrm d0$ nor $\mathrm dg(0)/\mathrm dx$.
Spivak in Calculus on Manifolds denotes the partial derivatives of $f$ as $D_1f,D_2f,\dots,D_nf$, and higher-order derivatives as $D_{1,1}f$ and so on. On pages 44-45 he discusses the drawbacks of the classical $\partial f/\partial x$ notation.
A: the whole problem of naming is something we can't completely erase. but we can avoid it using conventions:
instead of $F_x(u,v)=2u$ we will say $F_u(u,v)=2u$.
for points we will use $x_0,x_1,\cdots,x_n$ and $y_0,y_1,\cdots,y_n$
and such.
conventions really are just a temporary solution until one start to understand from how the it was written because they are not the same everywhere.
for example i saw times where a function with $2$ variables look like this: $F(x_0,x_1)$, i.e. $x_0,x_1$ are the variables, and saw other texts writing $\text{"at the point $(x_0,y_0)$..."}$, in this paper it is clear that $x_0$ is not a refer to variable.

the notation @spaceisdarkgreen suggested is a nice way but this notation can create even more confusion when you work with sequences of functions.
it is a lot more effective to write what you want to do clearly.

although that i said that conventions are not always the same some of them are so widely used so even if you don't use the convention you will understand it.
@Doris suggested a nice way
another ways to write it: 
$$\partial_{n_1}^{k_1}\partial_{n_2}^{k_2}\partial_{n_3}^{k_3}\cdots f$$ 
and
$$\frac {\partial ^{k_1+k_2+k_3+\cdots}f}{\partial {n_1}^{k_1}\,\partial {n_2}^{k_2}\,\partial {n_3}^{k_3}\cdots}$$
where $k_i$ the how many derivative we take for the variable $n_i$
other way is:$$D_{a_1,a_2,a_3,\cdots}f$$
here $a_i$ is really the place of the variable(e.g. $F_x(x,y)=D_1F(x,y)$)
here for higher degree of derivative you repeat the same number: $D_{1,1,2}f$ is the second derivative of the first variable and the first derivative of the second variable.
