Test Functions on Euclidean Space, Their Gradient Maps and Laplacians

Adopt Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation and fix a dimension $q$. The set $\mathbb{R}^{q}$ is open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, and its open subsets form a topology by Metric Open Sets Form a Topology and Euclidean Openness Agrees with Metric Openness on $\mathbb{R}^n$.

1. (Test functions)¶ A test function on $\mathbb{R}^{q}$ is a function $\psi:\mathbb{R}^{q}\to\mathbb{R}$ that is smooth on $\mathbb{R}^{q}$ and compactly supported with respect to that topology. The set of all test functions on $\mathbb{R}^{q}$ is denoted $C_{c}^{\infty}(\mathbb{R}^{q})$.

2. (Gradient map and Laplacian)¶ Let $\psi\in C_{c}^{\infty}(\mathbb{R}^{q})$. Being smooth, $\psi$ is of class $C^{k}$ on $\mathbb{R}^{q}$ for every natural number $k$, in particular of class $C^{1}$ and of class $C^{2}$, in the sense of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §derivatives. The gradient map of $\psi$ is $\nabla\psi:\mathbb{R}^{q}\to\mathbb{R}^{q}$, $x\mapsto D\psi(x)$, where $D\psi(x)$ is the gradient of $\psi$ at $x$, the point of $\mathbb{R}^{q}$ whose $i$th coordinate is the partial derivative $\partial_{i}\psi(x)$; and $\Delta\psi:\mathbb{R}^{q}\to\mathbb{R}$ is the Laplacian of $\psi$.