Theory of drawing (multivariable) functions (I am new)

Question

My question is a bit different. When is it possible to draw a function? E.g a 4-dimensional vector space is no more imaginable. If we a function from $\mathbb R^2$ to $\mathbb R$, we can draw it 3- dimensional. If we use polar coordinates we could at least draw the codomain in $\mathbb R^2$. So whats the theory behind it ? Until which point can i really visualize it? and when is it no more possible, would appreciate a long answer or at least link to an explanation

Answer 1

A function $f$ from $X$ to $Y$ should for these purposes be thought of as a subset of the Cartesian product of $X$ and $Y$ - that is, if we can first picture $X\times Y$ we can picture $f$ by thinking of it as a "shape" living in $X\times Y$.

The crucial obstacle now is the dimension. If $X=\mathbb{R}^m$ and $Y=\mathbb{R}^n$, then $X\times Y$ is basically the same as $\mathbb{R}^{m+n}$ - this is why trying to graph a function from $\mathbb{R}^2$ to $\mathbb{R}^2$ seems to force one into four-dimensional space.

Changing coordinate systems doesn't really help, unless you adopt a truly evil coordinate system (in which case your picture is ruined anyways): for example, even the shift from rectangular to polar coordinates (contrary to what you seem to claim) keeps the dimension of the plane the same (we still use two numbers to describe a given point). Once $X$ and $Y$ have some reasonable notion of geometry, we've really determined what $X\times Y$ is going to look like, and in particular how complicated it will be. Switching from one "reasonable description" to another won't result in a drastic change, although it may improve some aspects of the picture (e.g. make it easier to figure out the bounds in some integral).

This is all a first step into topology, and more specifically the study of manifolds.