On multivariable functions and graphs

Question

The definition of graph from Wikipedia is the following:

In mathematics, the graph of a function f is, formally, the set of all ordered pairs $(x, f(x))$, and, in practice, the graphical representation of this set. If the function input $x$ is a real number, the graph is a two-dimensional graph, and, for a continuous function, is a curve. If the function input x is an ordered pair $(x_1, x_2)$ of real numbers, the graph is the collection of all ordered triples $(x_1, x_2, f(x_1, x_2))$, and for a continuous function is a surface.

So for a function $f: \mathbb{R}^n \rightarrow \mathbb{R}$, its graph would be the set $\{x_1,...,x_n,f(x_1,...,x_n)\}$.

• When I plot the graph of the function $f(x,y,z)=x+y+z$ on Wolfram alpha, the following picture shows. My question is, what does this represent? Isn't its graph $\{x,y,z,f(x,y,z)\}$ and therefore 4-dimensional and not possible to plot in 3 dimensions?

• I haven' t found information about graphs of functions with values in $\mathbb{R}^m$. If we consider a function $f': \mathbb{R}^n \rightarrow \mathbb{R}^m$, although its graphical representation would be a mess, would it make sense to define its graph as the set $\{x_1,...,x_n,f_1(x_1,...,x_n),f_2(x_1,...,x_n),...,f_m(x_1,...,x_n)\}$?

Answer 1

The definition of graph is exactly the one you have given, for a function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ the graph is indeed the set $\{x_1,...,x_n,f(x_1,...,x_n)\}$.

For $n=3$ we cannot plot the graph for the whole function, what we can do is to plot the graph of the level surfaces $f(x,y,z)=k$ (maybe those are the surfaces shown by wolfram in your example).

For the general case $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ the definition of graph could be done for each single component in $\mathbb{R}^m$.

Answer 2

1. I have not checked, but those might be the level surfaces of the function of $3$ variables. In principle, any function $$f:\mathbb R^n\to \mathbb R$$ can eventually be reduced to a collection of level curves, namely curves in $\mathbb R^2$.

2. Functions $$f':\mathbb R^n\to \mathbb R^m$$ are usually thought of as euclidean vector fields (the vectors having dimension $m$) in $\mathbb R^n$.