Image pulled from the textbook: http://puu.sh/BS7Ca/7d4f9697e2.png
I understand the definition of a partial derivative by limit, and I also understand the method of computing partial derivatives. However, I am confused about what the textbook is trying to tell me. I do not get the part where it says "we restrict $f$ to the line $y=x...$" and everything after. If someone could explain to me what the restriction means I think I could probably understand the rest. My intuition of the restriction is that only values where $y=x$ are taken on, but that doesn't make sense because the graph of $x^{1/3}$ and $x^{2/3}$ only have values $y=x$ at $x = 0$ and $x = 1...$
Note that the graph of $z=f(x,y)$ is a surface compromised of all $$ (x,y,z)$$ such that $z=x^{1/3}y^{1/3}$
On the other hand $x=y$ in three dimensional space is a vertical plane passing through the line $y=x$
The intersection of this vertical plane and the graph of $z=x^{1/3}y^{1/3}$ is a curve which is compromised of all $(x,x,x^{2/3})$ where x is a real number. This curve is the restriction of $z=x^{1/3}y^{1/3}$ with $y=x$.