Laplace operator in spherical coordinates

We adopt the standard convention from physics, whereby $\theta$ is the polar angle (also referred to as co-elevation) and $\phi$ is the azimuthal angle, as shown in the figure below:

We denote by $(\underline{a}_{\langle\vartheta\rangle},\underline{a}_{\langle\phi\rangle})$ the corresponding orthonormal basis, which we show in the following sketch:

We find it convenient to think of $\vartheta$ and $\phi$ as scalar fields defined over the sphere. A direct calculation yields the following relations:

The adoption of $\underline a^\vartheta$ and $\underline a^\phi$ turns into a particularly handy notation when expressing the gradient of a scalar field $Y$:

Here we associate a the numerical indexes $1$ and $2$, respectively, to the coordinates $\vartheta$ and $\phi$, and we use a comma to denote partial differentiation, so that, given a scalar field $Y$, the symbols $Y_{,1}$ and $Y_{,2}$ denote the partial derivatives of $Y$ with respect to $\vartheta$ and $\phi$, respectively; moreover, when performing summations we adopt Einstein convention, with Greek letters for dummy indexes that run over the set ${1,2}$.

The Laplacian of a vector field is then:

By observing that $\underline{a}_{\langle\alpha\rangle, \beta} \cdot \underline{a}_{\langle\alpha\rangle}=0$ and that $\underline{a}_{\langle\theta\rangle,\phi}=(\cos \vartheta) \underline{a}_{\langle\phi\rangle}$, we compute