How to define a manifold for representing PDEs in vectorial form?

asked 2023-12-03 04:23:20 +0100

Hello folks, I need some help, I want to be able to represent equations like $\rho \frac{\partial^2 \mathbf u(x,y,z,t)}{\partial t^2} - c^2 \nabla^2 \mathbf u(x,y,z,t) = 0$, in which u is a vector field, but in this case $\nabla^2 \mathbf u = \frac{\partial^2\,u_{x}}{\partial x ^ 2}e_x + \frac{\partial^2\,u_{y}}{\partial y ^ 2}e_y + \frac{\partial^2\,u_{z}}{\partial z ^ 2}e_x$ is actually a Laplacian only in the physical dimensions x, y, z, not time. As a consequence, the gradient, divergent and curl should only be applied to the physical dimensions as well. I wasn't able to reproduce this behavior for a 4D Euclidean space field:

from sage.manifolds.operators import *

E.<x,y,z,t> = EuclideanSpace()

u = E.scalar_field(function('u')(x,y,z,t))


E^4 → ℝ

(x, y, z, t) ↦ d^2(u)/dx^2 + d^2(u)/dy^2 + d^2(u)/dz^2 + d^2(u)/dt^2

I suspect the way to solve is by creating a manifold with a custom frame, but I have no idea of where to begin since my knowledge of manifolds is minimal. Any thoughts on how I can do this, thanks!

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