Today, we talk about Laplacian operator.

Take $\R^n$ to be our space. Let $x_1, \cdots, x_n$ be the standard coordinates. Let $u_1, \cdots, u_n$ be a general curvilinear coordinates. Then, we have vector basis transformation rule $$ \frac{\d}{\d u_i} = \sum_j \frac{\d x_j}{\d u_i}\frac{\d}{\d x_i}. $$

We can then compute the component of metric tensor in $u_i$ coordinates $$g_{ij} = g (\frac{\d}{\d u_i}, \frac{\d}{\d u_j}) = \sum_k \frac{\d x_k}{\d u_i} \frac{\d x_k}{\d u_j} $$

If we view $g_{ij}$ as entries of a matrix $[g_{ij}]$, then we denote $g^{ij}$ the entries of the inverse matrix. Let $|g| = \det( [g_{ij}] )$ be the absolute value of the determinant of the matrix. Then, our formula for the Laplacian operator is (where $\d_i = \d / \d u_i$)

$$ \Delta f = \frac{1}{\sqrt{|g|}} \d_i (g^{ij} \sqrt{|g|} \d_j(f)) $$