= Heat invariants
{title2=$\operatorname{Tr}e^{-t\Delta}\sim(4\pi t)^{-d/2}\sum_{j\geq0}a_jt^j$}
For the positive scalar <Laplace-Beltrami operator> on a compact manifold without boundary, the heat invariants are the integrated coefficients $a_j=\int_Mu_j(x,x)\,dV_g$ in the short-time <heat kernel expansion>. In particular $a_0=\operatorname{Vol}(M)$ and $a_1=\tfrac16\int_M R_g\,dV_g$, with $R_g$ the <scalar curvature>. They are determined by the <heat trace>, hence by the <spectrum> with multiplicities. The leading positive coefficient recovers the dimension from $\lim_{t\downarrow0}\log\operatorname{Tr}(e^{-t\Delta})/\log(1/t)=d/2$.
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