Does the Laplace operator include the second derivative with respect to time variable?

Question

Does the Laplacian of a function $f(x,z,t)$ equal $f_{xx} + f_{yy} + f_{tt}$?

We aren't sure whether or not time is included in it or not.

Answer 1

It depends... the Laplacian is always defined with respect to some metric: $\Delta f = \nabla \cdot \nabla f$ and divergence requires a metric. Alternatively, you can define $\Delta$ as the gradient, in the sense of the calculus of variations, of the Dirichlet energy $\int \langle \nabla f, \nabla f\rangle\,dV$ and here again the metric is seen.

My guess is that for classical fluid dynamics the metric is simply the Euclidean $dx^2+dy^2+dz^2$. In which case $\Delta f(x,z,t) = f_{xx} + 0 + f_{zz}.$