A Note on the Wodzicki Residue

In this note we explain the relationship of the Wodzicki residue of (certain powers of) an elliptic differential operator $P$\ acting on sections of a complex vector bundle $E$\ over a closed compact manifold $M$\ and the asymptotic expansion of the trace of the corresponding heat operator $e^{-tP}$. In the special case of a generalized laplacian $\mathrm{△}$\ and ${{\rm dim};M

2}$, we thereby obtain a simple proof of the fact already shown in [KW], that the Wodzicki residue $res({\mathrm{△}}^{-\frac{n}{2}+1})$\ is the integral of the second coefficient of the heat kernel expansion of $\mathrm{△}$\ up to a proportional factor.