= Heat-kernel transport equations
{title2=$(r\partial_r+i+\tfrac r2\partial_r\log J)w_i=\Delta w_{i-1}$}
In <normal coordinates> $q=\exp_p x$, write the <Riemannian volume form> as $J(p,x)\,dx$ and let $r=|x|$. With the <heat operator> $\partial_t-\Delta_q$, the Gaussian times a power-series amplitude has its singular terms cancelled by the displayed radial equations for $i\geq1$. The initial coefficient is $w_0=J^{-1/2}$, and
$$
w_i(p,\exp_p x)=J(p,x)^{-1/2}\int_0^1s^{i-1}J(p,sx)^{1/2}(\Delta_qw_{i-1})(p,\exp_p(sx))\,ds.
$$
Multiplying the equation by $r^{i-1}J^{1/2}$ gives the derivative of $r^iJ^{1/2}w_i$. Smoothness forces the integration constant to vanish. The integral on a fixed compact interval proves smoothness across the diagonal by induction; it yields $w_i(p,p)=\Delta_qw_{i-1}(p,p)/i$. A truncated <heat parametrix> therefore has residual $-g t^k\Delta_qw_k$.
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