  In this paper we extend a gradient estimate of R. Hamilton for positive
solutions to the heat equation on closed manifolds to bounded positive
solutions on complete, non-compact manifolds with $Rc \geq -Kg$. We accomplish
this extension via a maximum principle of L. Karp and P. Li and a
Bernstein-type estimate on the gradient of the solution. An application of our
result, together with the bounds of P. Li and S.T. Yau, yields an estimate on
the gradient of the heat kernel for complete manifolds with non-negative Ricci
curvature that is sharp in the order of $t$ for the heat kernel on
${\mathbb{R}}^n$.
