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In this short paper, we derive an alternative proof for some known [van denBerg & Gilkey 2015] short-time asymptotics of the heat content in compactfull-dimensional submanifolds $S$ with smooth boundary. This includes formulaelike \begin{equation*} \int_{S} \exp(t\Delta)\left( f \mathbb 1_S\right)\,\mathrm{d}x = \int_S f \,\mathrm{d}x - \sqrt{\frac{t}{\pi}} \int_{\partial S} f\,\mathrm{d}A + o(\sqrt t),\quad t \rightarrow 0\,. \end{equation*} and(partially new) explicit expressions for similar expansions involving otherpowers of $\sqrt t$. By the same method, we also obtain short-time asymptoticsof $\int_S \exp(t^m\Delta^m)\left(f \mathbb 1_S\right)\, \mathrm{d}x$, $m \in\mathbb N$, and more generally for one-parameter families of operators $t\mapsto k(\sqrt{-t\Delta})$ defined by an even Schwartz function $k$.

Short-time heat content asymptotics via the wave and eikonal equations