Consider the following surfaces S1: xyz=10 and S2: z=x^2+y^2
I cannot understand whether S1 and S2 are three dimensional or 4 dimensional.
S1 seems to be 4 dimensional, as I can consider x, y and z as independent variables. S2 seems to be 3 dimensional, as z is dependent on x and y.
However, my professor wrote G(x,y,z)=xyz and H(x,y,z)=x^2+y^2-z while solving a problem related to these surfaces.
Don't the functions G and H show that x,y,z are independent variables, so the function is in 4D?
I am very confused, please help :(
I'll expand on my comment.
Let's be a little more careful about what $S_1$ and $S_2$ are. Let's start with $S_1$ $$S_1=\{(x,y,z)\in\Bbb{R}^3 : xyz=10\}$$
So we can see that $S_1$ is a subset of 3-dimensional space ($\Bbb{R}^3)$ cut out by a single equation/condition $xyz=10$. Thus we have three independent variables on which we impose one constraint, so $S_1$ is a 2-dimensional subset of 3-dimensional space.
Similarly $$S_2=\{(x,y,z)\in\Bbb{R}^3:z=x^2+y^2\}$$ is also a subset of 3-dimensional space cut out by a single equation, so $S_2$ is 2-dimensional.
The confusion you seem to be having with respect to the functions is that $x$, $y$, and $z$ are independent variables in $\Bbb{R}^3$, but not on $S_1$ or $S_2$ (since we imposed constraints on the variables to define the surfaces).
Not sure how you got to 4-dimensional space, or four independent variables though.