Confusion about $\Delta_{S^n} u(x) = \Delta u\bigg(\frac{x}{|x|}\bigg)$, why do we need to divide by $|x|$?

Question

Let $S$ be a sphere of radius 1. We know the formula $$\Delta_S u(x) = \Delta u\bigg(\frac{x}{|x|}\bigg)$$ holds for a function $u:S \to \mathbb{R}$ where $\Delta_S$ is the Laplace-Beltrami and the $\Delta$ is the usual Laplacian.

I am a bit confused about this formula. We only really want to use $\Delta_S u(x)$ for points $x \in S$, in which case $|x|=1$, so why can't we just write $$\Delta_S u(x) = \Delta u(x)?$$

Why bother with the division?

Answer 1

If $u$ is defined only on $S$ then $\Delta u$ doesn't make sense doesn't it?

We are not evaluating "$\Delta u$" at $x/|x|$, we are taking $\Delta$ of the function $u(x/|x|)$ (then plugging in).