TITLE: Some basic doubts about partial differentiation - PART I
QUESTION [15 upvotes]: EDIT : I had earlier asked four doubts together and I thought that maybe I should ask some of them in a separate question.
I have some very basics doubts about partial differentials ....
Doubt #1 :  What is the correct mathematical interpretation of the statement "$x$ and $y$ are independent variables" ? Does it mean $\large \frac {\partial x}{\partial y}=\frac{\partial y}{\partial x}=0$ ? According to my understanding of independent variables, if $x$ and $y$ are given to be independent, then any change in $y$ and $y$ only should not cause any change in $x$. So $ \large \frac{\partial x}{\partial y}$ must be zero. Similar logic for $\large \frac {\partial y}{\partial x} =0$. Is it okay to use this logic ?
Doubt #2 : : Under what circumstances is $\large\frac{\partial x }{\partial y} = \Large \frac{1}{\frac{\partial y}{\partial x}}$ valid?
These doubts have become a real hinderance to my understanding of partial derivatives. I need some insightful examples so that I can understand the subtleties that I am currently overlooking.
Rest of the doubts can be found here : Some basic doubts about partial differentiation - PART II

REPLY [3 votes]: To address the first question, I think the answer is that there is no formal definition of what $x$ and $y$ being independent variables means. What one has in mind when one says that $x,y$ are independent is that if we know $x$, we do not necessarily know $y$, and vice versa: in other words $x,y$ can take on different values "independently" of the other. So for instance, say we wanted to find the minima and maxima of
$$f(x,y)=\sin\left( x^2\right)+\cos(xy^2)$$
we can say that $x,y$ are independent, but if we were to do so under the additional restriction that $x=y^2$, then they are no longer independent because knowing the value of $x$ tells us the value of $y$ and vice versa.
You also want to take note that it doesn't really make much sense to make a statement like
$$\frac{\partial x}{\partial y}=\frac{\partial y}{\partial x}=0.$$
A notation like $\frac{\partial f}{\partial x}$ means taking the partial derivative of $f$ with respect to $x$, but here what are we taking the partial derivatives of? A constraint like the example $x=y^2$ given above? In that case, it is certainly not true: as long as we have such a constraint, then the variables are already not said to be independent.