Let $B$ be an area with smooth boundary, $U$ and open superset of $B$.
Let $F:(p,t) \in U \times \mathbb{R} \mapsto F(p,t) \in \mathbb{R}$ be a function that is two times continuously differentiable in the three variables $p = (x,y,z)$ und a time variable $t$.
We define $\Delta _p F = \partial_x ^2 F + \partial_y ^2 F + \partial_z ^2 F$
Lets assume $F$ meets all the criteria for the wave equation:
- $\partial_t^2F (p,t) = \Delta_p F(p,t) $ for all $(p,t)\in B \times\mathbb{R}$
- $F(p,t) = 0$ for all $(p,t)\in \partial B \times\mathbb{R}$
Show that the energy function $E:\mathbb{R}\to \mathbb{R}_{\geq 0}$ defined by $$E(t) = \int_{B}((\partial_t F(p,t))^2 +(\partial_x F(p,t))^2+(\partial_y F(p,t))^2+(\partial_z F(p,t))^2)dvol(p)$$ is constant for $t\in \mathbb{R}$
The hints given to solve this problem are:
either assume that $B=Q$ is a cuboid or use (without proving it) multidimensional differentiation under the integral.
How I tried solving this
If $E(t)$ is constant over time we know $\frac{\partial E}{\partial t} = 0$ $$\frac{\partial E}{\partial t} = \frac{\partial }{\partial t}\int_{B}((\partial_t F(p,t))^2 +(\partial_x F(p,t))^2+(\partial_y F(p,t))^2+(\partial_z F(p,t))^2)dvol(p)\\ = \int_{B}\frac{\partial }{\partial t}\left(((\partial_t F(p,t))^2 +(\partial_x F(p,t))^2+(\partial_y F(p,t))^2+(\partial_z F(p,t))^2) \right)dvol(p)$$
Since all points in the intergral are in $B$ we can use the fact $\partial_tF (p,t) = \Delta_p F(p,t) $ for all $(p,t)\in B \times\mathbb{R}$ from above giving me $$\int_{B}\frac{\partial }{\partial t}\left(((\partial_x ^2 F + \partial_y ^2 F + \partial_z ^2 F)^2 +(\partial_x F)^2+(\partial_y F)^2+(\partial_z F)^2) \right)dvol(p)$$
Somehow I must show that this is equals to $0$ I am pretty lost so I really appreciate your help.
Since $B$ has a smooth boundary and the hint tells me to look at $B$ as a cuboid I assume on can use Stokes Theorem or Gauss Theorem to solve this.