I got stuck on a formula in a proof on the mean value property of harmonic functions in a sphere In looking over a proof on the mean value property for harmonic functions on spheres I came across the following formula:
$$\frac{\partial u}{\partial n}=\textbf{n}\cdot \triangledown u=\frac{\bf{x}}{r}\cdot \triangledown u =\frac{x}{r}u_x+\frac{y}{r}u_y+\frac{z}{r}u_z=\frac{\partial u}{\partial r}$$
Where $r=\sqrt{x^2+y^2+z^2}$ is the spherical distance to the center of the circle and because we are considering a sphere $\textbf{n}=\frac{\bf{x}}{r}$ is the outward pointing unit normal. I can understand everything until the last equality sign. I would be inclined to think that 
$$\frac{\partial u}{\partial r}=u_r=u_x\frac{\partial x}{\partial r}+u_y\frac{\partial y}{\partial r}+u_z\frac{\partial z}{\partial r}$$
Now $r=\sqrt{x^2+y^2+z^2}$ implies that $x=\sqrt{r^2-y^2-z^2}$ and thus:
$$\frac{\partial x}{\partial r}= \frac{2r}{2\sqrt{r^2-y^2-z^2}}=\frac{r}{x}  $$
Similarly for $y$ and $z$ leading to the conclusion that
$$u_r=\frac{r}{x}u_x+\frac{r}{y}u_y+\frac{r}{z}u_z$$
Which has all the fractions multiplying the partial derivatives inversed. I must be missing something obvious here or be making a stupid mistake, if anybody would point it out to me that would be very nice. I want to understand this proof and I can't continue if my thoughts completely disagree with what it says because that way I can never have confidence in the result!
 A: This is a very common type of mistake that's caused by our suboptimal notation for partial derivatives. See this answer for a collection of answers I've written in response to similar confusions over partial derivatives.
The symbol $\displaystyle\frac{\partial x}{\partial r}$ in the context in which you're using it is intended to mean $\displaystyle\left.\frac{\partial x}{\partial r}\right|_{\theta,\phi}$, that is, the rate of change of $x$ with $r$ as you hold the angular coordinates $\theta$ and $\phi$ of the spherical coordinates $r$, $\theta$ and $\phi$ constant. What you've calculated is $\displaystyle\left.\frac{\partial x}{\partial r}\right|_{y,z}$, that is, the rate of change of $x$ with $r$ as you hold the Cartesian coordinates $y$ and $z$ constant. Such confusion arises very easily because our standard notation for partial derivatives doesn't show what's being held constant and is therefore ambiguous.
With $x=r\sin\theta\sin\phi$, you immediately find $\displaystyle\left.\frac{\partial x}{\partial r}\right|_{\theta,\phi}=\sin\theta\sin\phi=\frac xr$ as expected.
