$\nabla ^2\psi$ equals $\psi -$ average value of $\psi$ at neighboring points Let $\psi (x,y,z)$ be a scalar field. I found the following statement in Morse & Feshbach Methods of Theoretical Physics:

The limiting value of the difference between $\psi$ at a point and the average value of $\psi$ at neighboring points is $-\frac{1}{6}(dxdydz)^2\nabla ^2\psi$

By taking a small sphere centered at the point of radius $r$, I was able to show that the difference between $\psi$ at a point and the average value of $\psi$ at neighboring points equals $-\frac{1}{6}r^2\nabla ^2\psi$. But I have not been able to get the expression given in the book. Thanks.
Edit: To obtain $-\frac{1}{6}r^2\nabla ^2\psi$ I first expanded $\psi$ in a Taylor series up to quadratic terms about the point in question. Then I integrated $\psi$ (rather, its Taylor expansion up to quadratic terms) over the surface of a small sphere of radius $r$ centered at the given point and divided this by the surface are of the sphere (this gives the average value of $\psi$ in the sphere). Subtracting this average from the value of $\psi$ at the center of the sphere gives the result $-\frac{1}{6}r^2\nabla ^2\psi$.
 A: I think this is a typo in Morse & Feshbach Methods of Theoretical Physics. The correct expression is $-\frac{1}{6}r^2\nabla ^2\psi$ or $-\frac{1}{6}(dx^2+dy^2+dz^2)\nabla^2 \psi$.
A: Formulated this way, $-\frac16\,r^2\,\nabla^2\psi$ is indeed the correct result. While it is certainly more elegant to express such things by means of differential forms rather than plain functions with finite $r$, the result as given in the book is not a meaningful differential form at all, since
$$
(\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z)^2
= \mathrm{d}x\wedge\mathrm{d}y\wedge\mathrm{d}z\wedge\mathrm{d}x\wedge\mathrm{d}y\wedge\mathrm{d}z
$$
$$
= (-1)(-1)\mathrm{d}x\wedge\mathrm{d}x\wedge\mathrm{d}y\wedge\mathrm{d}z\wedge\mathrm{d}y\wedge\mathrm{d}z = 0,
$$
due to $\mathrm{d}x_i\wedge\mathrm{d}x_j=-\mathrm{d}x_j\wedge\mathrm{d}x_i$. The correct way might be something like
$$
-\frac16\,(\mathrm{d}y\mathrm{d}z+\mathrm{d}z\mathrm{d}x+\mathrm{d}x\mathrm{d}y)\,\nabla^2\psi
$$
but I'm not sure if this would much more meaningful. The plain finite-small-$r$ expression is probably not the worst after all.
