$\phi(\mathbf{r}) = \log r$ のとき、$\frac{\partial \phi}{\partial x}$, $\frac{\partial \phi}{\partial y}$, $\frac{\partial \phi}{\partial z}$ を求める問題です。ここで、$r = \sqrt{x^2 + y^2 + z^2}$です。

1. 問題の内容

$\phi(\mathbf{r}) = \log r$ のとき、$\frac{\partial \phi}{\partial x}$, $\frac{\partial \phi}{\partial y}$, $\frac{\partial \phi}{\partial z}$ を求める問題です。ここで、$r = \sqrt{x^2 + y^2 + z^2}$です。

2. 解き方の手順

まず、$r = \sqrt{x^2 + y^2 + z^2}$から$\frac{\partial r}{\partial x}$, $\frac{\partial r}{\partial y}$, $\frac{\partial r}{\partial z}$を求めます。

$\frac{\partial r}{\partial x} = \frac{1}{2\sqrt{x^2 + y^2 + z^2}} \cdot 2x = \frac{x}{\sqrt{x^2 + y^2 + z^2}} = \frac{x}{r}$

同様に、

$\frac{\partial r}{\partial y} = \frac{y}{r}$

$\frac{\partial r}{\partial z} = \frac{z}{r}$

次に、$\frac{\partial \phi}{\partial x}$, $\frac{\partial \phi}{\partial y}$, $\frac{\partial \phi}{\partial z}$を求めます。

$\phi = \log r$なので、$\frac{\partial \phi}{\partial r} = \frac{1}{r}$

$\frac{\partial \phi}{\partial x} = \frac{\partial \phi}{\partial r} \cdot \frac{\partial r}{\partial x} = \frac{1}{r} \cdot \frac{x}{r} = \frac{x}{r^2}$

$\frac{\partial \phi}{\partial y} = \frac{\partial \phi}{\partial r} \cdot \frac{\partial r}{\partial y} = \frac{1}{r} \cdot \frac{y}{r} = \frac{y}{r^2}$

$\frac{\partial \phi}{\partial z} = \frac{\partial \phi}{\partial r} \cdot \frac{\partial r}{\partial z} = \frac{1}{r} \cdot \frac{z}{r} = \frac{z}{r^2}$

3. 最終的な答え

$\frac{\partial \phi}{\partial x} = \frac{x}{r^2}$

$\frac{\partial \phi}{\partial y} = \frac{y}{r^2}$

$\frac{\partial \phi}{\partial z} = \frac{z}{r^2}$