Let $f: M \rightarrow \mathbb{R}^3$ be an immersion of a surface $M$. For pedagogical purposes (i.e., I'm teaching a class!) I am looking for an expression for the scalar Laplace-Beltrami operator $\Delta$ applied to a real function $\phi$ on $f(M)$ that:

- explicitly depends on the immersion $f$,
- does not rely on local coordinates, and
- does not use exterior calculus.

A standard *coordinate* expression is

$$\Delta \phi = \frac{1}{\sqrt{g}} \partial_i (\sqrt{|g|} g^{ij} \partial_j \phi),$$

and a standard expression using exterior calculus is

$$\Delta\phi = \star d \star d \phi.$$

However, the students do not have exposure to exterior calculus, and I am discouraging the use of coordinates whenever possible (and have so far been able to get by without them).

To give a concrete example of the "style" of expression I'm looking for, consider the normal curvature in a direction $X \in TM$, which can be expressed as

$$\kappa_n(X) = -\frac{dN(X) \cdot df(X)}{|df(X)|^2},$$

where $N: M \rightarrow S^2 \subset \mathbb{R}^3$ is the Gauss map and $\cdot$ denotes the usual Euclidean inner product. This expression uses the differential $d$ of a function, but it does not use the exterior derivative on $k$-forms (at least, not for $k>0$), nor does it use the Hodge star, nor does it rely on a coordinate system.

In English, $\Delta$ is not hard to describe: take the sum of second derivatives along orthogonal directions in the ambient space. But after a lot of digging, I'm surprised to find there isn't a more suggestive algebraic description.

Thanks!