Explicit example of $f\in\mathcal{D}(\mathbb{R}^{3})$

Question

I want to use a specific example of a three dimensional test function, i.e. $f\in\mathcal{D}(\mathbb{R}^{3})$. Certainly, Wikipedia has an example of the one dimensional case but for the three dimensional case I can find absolutely nothing.

Answer 1

Take any $f\in \mathcal D (\mathbb R^1).$ Then $f(x_1^2+x_2^2+x_3^2) \in \mathcal D (\mathbb R^3).$

Answer 2

In fact, let $\Omega$ be a non-empty open subset of $\Bbb R^n$. Then there exists some $B(a,2\delta)\subset \Omega$. The function:

$$ \begin{cases} \exp\left(-\frac1{\delta^2 - \|x-a\|^2} \right), \text{ if $x\in B(a,\delta)$} \\ 0, \text{ otherwise} \end{cases}$$

is in $D(\Omega)$.