  In this short paper, we re-derive the Bochner formula for the Laplacian by
considering local variations of volume.
  The derivation is rooted in the fact that the Laplacian of a function
measures the volume variation along the flow of the gradient vector of the
function. Possible extensions of this approach/technique are also discussed.
While the value of this approach may be limited in terms of research, we think
it definitely has pedagogical value.
