equation about the Riemann metric and integral

Question

Let $f:\mathbb{R}^3\rightarrow \mathbb{R}$ be a $C^{\infty}$function and there exist a natural number $k$ such that $$ f(tx,ty,tz)=t^kf(x,y,z) $$ for all $t\in \mathbb{R}$. Let $\omega$ be the volume element on $S^2$ induced by the Riemann metric on $\mathbb{R}^3$. I want to show the following equation holds. $$ \int_{S^2}kf\omega=\int_{D^3}\left(\frac{\partial^2f}{\partial x^2}+\frac{\partial^2f}{\partial y^2}+\frac{\partial^2f}{\partial z^2}\right)dx\wedge dy\wedge dz$$ where $D^3$ is the unit disk on $\mathbb{R}^3$. First I got the following equation by differentiate the above equation by $t$. $$ \left(x\frac{\partial}{\partial x}+y\frac{\partial}{\partial y}+z\frac{\partial}{\partial z}\right)f=kf $$ Then I computed $$\int_{S^2}kf\omega$$ by using the Stokes' theorem and polar coordinates, but I was not able to get what I wanted. I'm still unfamiliar with differential forms, so there may be good ways. Please give me some advice.