My question concerns how the second-order term in a Taylor expansion of a scalar-valued function of a vector can be represented by a Laplacian instead of the product of vectors times dels. The origin of this question has led me to expect the numerical coefficient on the Laplacian term will have a denominator divisible by d, the dimension of the vector space. That is because the expansion below (with some assumptions) generates a differential equation solved by a Gaussian probability distribution with a variance scaled by d.
Thomas Witten in his book "Structured Fluids" looks at an unspecified function f of the magnitude of an $\mathbb{R^3}$ vector. He Taylor expands this function to second order to derive a differential equation. Here's a simplified version of his statement, and it's the second order term that's making me curious.
$p(|\vec{r}-\vec{r}_1|) = p(|\vec{r}|)-\vec{r}_1\cdot \nabla p(|\vec{r}|) + \frac{1}{6}r_1^2\nabla^2p(|\vec{r}|) $ + ...
I typically write my Taylor expansions as $f(\vec{x}-\vec{a}) = f(\vec{x}) - \vec{a}\cdot\nabla f(\vec{x}) + \frac{1}{2}(\vec{a}\cdot\nabla) (\vec{a}\cdot\nabla) f(\vec{x}) + ...$
How can I transform from the general second-order Taylor expansion to the quoted expansion with a Laplacian and different numerical prefactor? What changes if we look at a vector argument in $\mathbb{R^d}$?