Based upon the appendix in [18], we outline a rigorous derivation of the Laplace-Beltrami
operator. We use the convention that indices appearing once in the upper position and
once in the lower position are considered dummy indices and are implicitly summed
over their range, while non-repeated indices are considered free to take any value.
Derivatives are also denoted using the lower-index comma notation, for example g_{ij,k}.
Covariant vectors such as the gradient are those which, under a change of coordinate
system, change under the same transformation in order to maintain coordinate system
invariance. In contrast, contravariant vectors, such as velocity, should remain fixed
under a coordinate transformation requiring that their components change under the
inverse of the transformation to maintain invariance. With this in mind, we now
construct the fundamental differential operators we require for a 2-dimensional manifold
embedded in a 3-dimensional space. In order to express these operators in curvilinear
coordinates we start by assuming that we have a smooth surface parametrization given
by

x(ξ^{1},ξ^{2}) := (x^{1}(ξ^{1},ξ^{2}),x^{2}(ξ^{1},ξ^{2}),x^{3}(ξ^{1},ξ^{2})). |

Next we will define the Jacobian of x as the tensor

J_{i}^{j} = |

where J_{i}^{j} can be viewed as a covariant surface vector (by fixing the upper index) or as a
contravariant space vector (by fixing the lower index). The surface metric tensor g_{ij} can be
defined in terms of the J_{i}^{j} as

g_{ij} = ∑
_{k=1}^{3}J_{
i}^{k}J_{
j}^{k}. | (10.1) |

which can be considered to transform a contravariant quantity to a covariant quantity.
Similarly the conjugate tensor g^{ij}, which does the reverse transformation, is given
by

g^{11} = g_{
22}∕g, g^{12} = g^{21} = -g_{
12}∕g, g^{22} = g_{
11}∕g, | (10.2) |

where g is the determinant of g_{ij}. The metric tensor and its conjugate satisfy the
condition

δ_{i}^{j} = g_{
ik}g^{jk} = |

To construct the divergence operator we will also need the derivative of g with respect to
components of the metric, g_{ij}. We know that g is invertible and from linear algebra we have
that the inverse of the metric (10.2) satisfies g^{-1} = ^{⊤}, where is the cofactor matrix of g.
Therefore ^{⊤} = gg^{-1}, or in components ^{ij} = g(g^{-1})_{ji}. Using Jacobi’s formula for the
derivative of a matrix determinant with respect its entries, and since g is invertible, the
derivative of the metric determinant is

= tr = ^{ij} = g(g^{-1})_{
ji} = gg^{ij}. | (10.3) |

The partial derivative of a tensor with respect to a manifold coordinate system is itself not a
tensor. In order to obtain a tensor, one has to use *covariant* derivative, defined below. The
covariant derivative of a contravariant vector is given by

∇_{k}a^{i} = a_{
,k}^{i} + a^{j}Γ_{
jk}^{i}. | (10.4) |

where Γ_{jk}^{i} are *Christoffel Symbols of the second kind*. The *Christoffel symbols of the first kind*
are defined by

Γ_{ijk} = . |

Here we note that Γ_{ijk} is symmetric in the first two indices. To obtain the Christoffel
symbols of the second kind we formally raise the last index using the conjugate
tensor,

Γ_{ij}^{l} = Γ_{
ijk}g^{kl} | (10.5) |

which retains the symmetry in the lower two indices. We can now express the derivatives of the metric tensor in terms of the Christoffel symbols as

g_{ij,k} = Γ_{ikj} + Γ_{jki} = g_{lj}Γ_{ik}^{l} + g_{
li}Γ_{jk}^{l}. |

We now define the divergence operator on the manifold, ∇⋅X = ∇_{k}X^{k}. Consider first the
derivative of the determinant of the metric tensor g with respect to the components of some
local coordinates system ξ^{1},ξ^{2}. We apply the chain rule, making use of the derivative of the
metric tensor with respect to components of the metric (10.3) and the relationship (10.5), to
get

= = gg^{ij}g_{
ij,k} = gg^{ij}(Γ_{
ikj} + Γ_{jki}) = g(Γ_{ik}^{i} + Γ_{
jk}^{j}) = 2gΓ_{
ik}^{i}. | (10.6) |

We can therefore express the Christoffel symbol Γ_{ik}^{i} in terms of this derivative as

Γ_{ik}^{i} = = . | (10.7) |

Finally, by substituting for Γ_{ik}^{i} in the expression for the divergence operator

∇_{k}X^{k} | = X_{
,k}^{k} + X^{i}Γ_{
ki}^{k} | ||

= X_{,k}^{k} + X^{k}Γ_{
ik}^{i} | |||

= X_{,k}^{k} + X^{k}()_{
,k} |

we can deduce a formula for divergence of a contravariant vector as

∇⋅X = ∇_{k}X^{k} = | (10.8) |

The covariant derivative (gradient) of a scalar on the manifold is identical to the partial
derivative, ∇_{k}ϕ = ϕ_{,k}. To derive the Laplacian operator we need the contravariant form of the
covariant gradient above which can be found by raising the index using the metric tensor,
giving

∇^{k}ϕ = g^{kj}ϕ_{
,j}, | (10.9) |

and substituting (10.9) for X^{k} in (10.8) to get the Laplacian operator on the manifold
as

Δ_{M}ϕ = _{,i}. | (10.10) |

We now extend the above operator to allow for anisotropic diffusion in the domain by
deriving an expression for the surface conductivity from the ambient conductivity. The
gradient of a surface function scaled by the ambient conductivity tensor ^{p} is given
by

^{p}f = g^{mp}J_{
m}^{l}σ_{
kl}J_{j}^{k}g^{ij}. | (10.11) |

The surface gradient is mapped to the ambient space through the Jacobian J_{j}^{k}, scaled by the
ambient conductivity, and mapped back to the surface through J_{m}^{l}. The anisotropic Laplacian
operator is given by

^{2}f = ∇_{
k}^{k}f = ∇_{k}^{ij}∇_{j}f | (10.12) |

Therefore the surface conductivity tensor can be computed using the Jacobian tensor and the inverse metric as

= g^{-1}JσJ^{⊤}g^{-1}. | (10.13) |

Anisotropic diffusion is important in many applications. In the ambient Euclidean space, this can be represented by a diffusivity tensor σ in the Laplacian operator as

Δ_{M} = ∇⋅ σ∇. |

On our manifold, we seek the generalisation of 10.10, in the form

_{M}ϕ = ∇_{j}_{i}^{j}∇^{i}ϕ. |

where the _{i}^{j} are entries in the surface diffusivity. For a contravariant surface vector a^{j} we can
find the associated space vector A^{i} as A^{i} = J_{j}^{i}a^{j}. Similarly if A_{i} is a covariant space vector,
then a_{j} = J_{j}^{i}A_{i} is a covariant surface vector. Using these we can construct the anisotropic
diffusivity tensor on the manifold by constraining the ambient diffusivity tensor σ to the
surface. The contravariant surface gradient ∇^{i}ϕ is mapped to the corresponding space vector,
which lies in the tangent plane to the surface. This is scaled by the ambient diffusivity and
then projected back to a covariant surface vector. Finally, we use the conjugate
metric to convert back to a contravariant form. The resulting surface Laplacian
is

_{M}ϕ = ∇_{m}g^{lm}J_{
l}^{k}σ_{
jk}J_{i}^{j}∇^{i}ϕ. |

Following on from this we deduce that

_{i}^{j} = g^{jm}J_{
m}^{l}σ_{
lk}J_{i}^{k}. |

It can be seen that in the case of isotropic diffusion that _{i}^{j} = δ_{j}^{i} ⇔ σ = I,

_{i}^{j} = g^{im}J_{
m}^{k}σ_{
lk}J_{j}^{l} = g^{im}J_{
m}^{k}J_{
j}^{k} = g^{im}g_{
mj} = δ_{j}^{i}. |