I'm reading Marsden Vector Calculus, and I'm wondering:
If you have some function $z=f(x,y)$ namely $z=x+y$. Here $z$ depends of two variables. But if you also have some other function $(x,y)=f(z)$. In this case, $x$ and $y$ are function of the same variable or parameter. So good so far, but my question is, I know for example for the first function you get a subset of $\mathbb R^2$ and the height of $z$ will depend on the value of the subset. But I can't imagine in $\mathbb R^3$ what would be the graph like.
Thanks in advance guys. If you know someone or somewhere I can ask these questions please let me know. I'm a student and I currently have more questions than answers.
The graph of $z=f(x,y)$ is some surface in 3-D space. As you wrote, the “height” of the surface above/below a point on the $x$-$y$ plane is the value of $f$ at that point.
If, on the other hand, you have $(x,y)=g(z)$, the resulting graph depends on how you interpret those variables. If they’re the same coordinates in $\mathbb R^3$ as above, then the graph is a curve that runs generally in the direction of the $z$-axis in the sense that the curve will have at most one intersection with a plane that’s perpendicular to the $z$-axis. For example, $(x,y)=(\cos z,\sin z)$ produces a helix about the $z$-axis; $(z+1,2z-3)$ is a straight line that passes through $(1,-3,0)$; and so on.
However, it’s more common to interpret $(x,y)=g(z)$ as a parameterization of a plane curve. In this case, $x$ and $y$ are the usual Cartesian coordinates of $\mathbb R^2$, but $z$ is a parameter that’s not represented directly by the graph. Instead, the curve is traced out as $z$ varies over its allowed values. To avoid confusion with a 3-D curve, the parameter usually has another name such as $t$ or $\lambda$. Interpreted as a parameterized plane curve, $(\cos z,\sin z)$ is the unit circle, possibly retraced many times, while $(z+1,2z-3)$ is a segment of the line $y=2x-5$.
Note that in the 3-D interpretation, it’s also more usual to present the curve as a vector of three functions instead of two, even when one of the functions is the identity. Thus, that helix would more usually be described by the parameterization $(\cos t,\sin t,t)$.