The Density of a Nondegenerate Gaussian Law

In the setting of Gaussian Random Vectors: Standing Notation, let $k\in\mathbb{N}$, let $m\in\mathbb{R}^{k}$, and let $\Sigma\in\mathcal{S}(k)$ be positive definite, with inverse $\Sigma^{-1}$ (Invertibility of Positive Definite Matrices (i)) and determinant $\det\Sigma>0$ (The Determinant of a Positive Definite Matrix Is Positive (i)). Let $\exp$ be the exponential function, $\pi$ the number of The Number Pi (i), $\sqrt{\cdot}$ the square root of a nonnegative real number ($2\pi>0$ by Quarter-Turn Identities and Periodicity of Sine and Cosine (i)), and $f:\mathbb{R}^{k}\to\mathbb{R}$ the function

$$f(x)=\frac{1}{(\sqrt{2\pi})^{k}\sqrt{\det\Sigma}}\exp\Bigl(-\tfrac{1}{2}\,(x-m)\cdot\Sigma^{-1}(x-m)\Bigr).$$

(i) The Gaussian density. The Gaussian law $N(m,\Sigma)$ ($\Sigma$ is positive semidefinite, Rules for Symmetric and Positive Semidefinite Matrices: Closure, Self-Adjointness, Polarization, Congruence and Cones (x)) is the measure with density $f$ with respect to $\lambda_{k}$:

$$N(m,\Sigma)(B)=\int_{B}f\,d\lambda_{k}\qquad(B\in\mathcal{B}(\mathbb{R}^{k})).$$