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Vector Calculus: An Introduction

Daniel Daners

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!
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