\begin{equation} I = \int_{\mathbb{R}^n} e^{-\langle x, Qx \rangle} \, dx \end{equation}

\begin{equation} Q = Q^\top > 0 \end{equation}

\begin{equation} \det(Q) = \prod_{k=1}^{n} \lambda_k \end{equation}

\begin{equation} I = \pi^{n/2} (\det Q)^{-1/2} \end{equation}

\begin{equation} \mathcal{F}{ e^{-\pi \langle x, Qx \rangle} } (\xi) = (\det Q)^{-1/2} e^{-\pi \langle \xi, Q^{-1}\xi \rangle} \end{equation}

\begin{equation} u(x,t) = \frac{1}{(4\pi t)^{n/2}} \int_{\mathbb{R}^n} e^{-\frac{|x-y|^2}{4t}} \phi(y) \, dy \end{equation}

\begin{equation} \partial_t u = \Delta u \end{equation}

\begin{equation} u(x,0) = \phi(x) \end{equation}

\begin{equation} \mathcal{J}[u] = \int_\Omega \left( |\nabla u|^2 + \lambda |u|^4 \right) \, dx \end{equation}

\begin{equation} \frac{\delta \mathcal{J}}{\delta u} = -\Delta u + 2 \lambda |u|^2 u = 0 \end{equation}

\begin{equation} -\Delta u + 2 \lambda |u|^2 u = 0 \end{equation}

\begin{equation} u(r) = \frac{1}{\sqrt{\lambda}} \, \mathrm{sech}(r) \end{equation}

\begin{equation} E = \int_{\mathbb{R}^n} \left( |\nabla u|^2 + \lambda |u|^4 \right) dx \end{equation}

\begin{equation} E = \frac{2^{n-1}}{\lambda^{n/2}} \int_0^\infty \mathrm{sech}^{2n}(r)\, dr \end{equation}

\begin{equation} \boxed{E = \frac{\sqrt{\pi}}{\lambda^{n/2}} \frac{\Gamma(n/2)}{\Gamma((n+1)/2)}} \end{equation}