A formula of Walczak is applied to two situations in differential geometry. Holomorphic distributions on Kähler manifolds are studied, and it is shown how the formula simplifies to a Bochner type formula, which is particularly useful in the study of integrable distributions. Then the Walczak formula is applied in the context of harmonic morphisms, where it provides a means of investigating the vertical Laplacian of the dilation. It is shown that, under some additional conditions on the map and the domain, the $p$-energy is infinite for $p$ sufficiently large.