Skip to main content Contents Index Prev Up Next \(\newcommand{\vect}[1]{\boldsymbol{#1}}
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Section 2.1 Graphs
Given a function from a set
\(D\subset\mathbb R^2\) with values in
\(\mathbb R\) we can plot its value on the
\(x_3\) -axis as shown in
Figure 2.1 . The resulting object is a surface, called the graph of
\(f\text{.}\) Formally, it is the set
\begin{equation*}
\graph(f):=\bigl\{\bigl(x_1,x_2,f(x_1,x_2)\bigr)
\mid (x_1,x_2)\in D\bigr\}
\end{equation*}
in \(\mathbb R^3\text{.}\)
Figure 2.1. To obtain the graph of \(f\) plot \(f(\vect x)\) above \(\vect x\in D\text{.}\) As an example consider the graph of
\(f(x,y):=5y(1-y^2)(1/2-x^2)-x\) in
Figure 2.2 .
Figure 2.2. Graph of \(f(x,y):=5y(1-y^2)(1/2-x^2)-x\text{.}\)
If \(D\subset\mathbb R^N\) we still call the set
\begin{equation*}
\graph(f)
:=\bigl\{\bigl(\vect x,f(\vect x)\bigr)\mid\vect x\in D\bigr\}
\subset\mathbb R^{N+1}
\end{equation*}
the graph of \(f\) over \(D\) , even if we cannot visualise it!
