The Standard Normal Densities in Closed Form

In the setting of Gaussian Random Vectors: Standing Notation, let $k\in\mathbb{N}$, let $\exp$ be the exponential function, $\pi$ the number of The Number Pi (i) and $\sqrt{2\pi}$ the square root of $2\pi>0$ (Quarter-Turn Identities and Periodicity of Sine and Cosine (i)).

(i) On the real line. The standard normal density is

$$\varphi(x)=\frac{1}{\sqrt{2\pi}}\exp(-x^{2}/2)\qquad(x\in\mathbb{R}).$$

(ii) On $\mathbb{R}^{k}$. The standard normal density on $\mathbb{R}^{k}$ is

$$\varphi_{k}(x)=\frac{1}{(\sqrt{2\pi})^{k}}\exp\bigl(-\lVert x\rVert^{2}/2\bigr)\qquad(x\in\mathbb{R}^{k}).$$