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I am learning the spherical means method to solve three-dimensional wave equations, but I am confused about it. Here is my understanding of the method, I will mark where I found confusing.

First, we define some spherical means function, i.e., for any given point $M(x_0,y_0,z_0)$, any given radius $r$, $\bar{\mu}(r,t)=\frac{1}{4\pi r^2}\iint_{S_{r}^{M}} u(x,y,z)dS$, where $S_{r}^{M}$ means the surface of a ball with center $M$ and radius $r$.

Second, we need to dig some properties of $\bar{\mu}$, so we integrate the wave equation on a given ball with center $M$ and radius $r$ $\iiint_{B_{r}^{M}}\mu_{tt}dxdydz=a^2\iiint_{B_{r}^{M}}(\mu_{xx}+\mu_{yy}+\mu_{zz})dxdydz$, by Gauss, we have

$$a^2\iiint_{B_{r}^{M}}(\mu_{xx}+\mu_{yy}+\mu_{zz})dxdydz=a^2\iint_{S_{r}^{M}}(\frac {\partial{u}}{\vec{n}})dS=a^2\iint_{S_{r}^{M}}(\mu_xcos\alpha+\mu_y cos \beta+\mu_z cos \gamma)dS$$

where $\vec{n}$ is the normal vector. Then it states that $a^2\iint_{S_{r}^{M}}(\frac {\partial{u}}{\vec{n}})dS=a^2\iint_{S_{r}^{M}}(\frac {\partial{u}}{r})dS$. I know we can use the spherical coordinate($\vec{n}=(sin\phi cos\theta,sin\phi cos\theta,cos\phi)$), and by the chain rule, it seems right($\mu_r=\mu_x x_r+\mu_y y_r+\mu_z z_r$). \begin{cases}x=x_0+rsin\phi cos\theta \\ y=y_0+rsin\phi cos\theta\\ z=z_0+rcos\phi \end{cases}, $r$ is given, $\phi \in [0,2\pi), \theta \in [0,\pi)$

But my question is, since we integrate on a given ball ($r$ is given), $\frac {\partial{u}}{r}$ is differentiating with respect to some constant. It does not make sense to me.

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  • $\begingroup$ It's a normal derivative, not differentiation w.r.t. a constant: $\frac{\partial u}{\partial n}=\nabla u\cdot n$ $\endgroup$ Nov 8, 2022 at 6:08

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I can already answer my own question. If we use $k$ instead of $r$ when integrating. It will be clearer and things just work out.

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