Example 4. Find the divergence of the vector

$$
\mathbf{a}=\varphi(r) \mathbf{r}^{0}=\frac{\varphi(r)}{r} \mathbf{r}
$$

where $r=|r|$ - the distance from the origin to the variable point $M(x, y, z)$.
Solution. Using formula (5), we get

$$
\mathrm{dlva}=\frac{\varphi(r)}{r} \operatorname{div} \mathrm{r}+\left(r, \operatorname{grad} \frac{\varphi(r)}{r}\right)
$$

Further,

$$
\text { div } r=3, \quad \text { grad } \frac{\varphi(r)}{r}=\left(\frac{\varphi(r)}{r}\right)^{\prime} \operatorname{grad} r=\frac{r \varphi^{\prime}(r)-\varphi(r)}{r^{2}} r^{0}
$$

Therefore,

$$
\operatorname{div} a=3 \frac{\varphi(r)}{r}+\left(\frac{r \varphi^{\prime}(r)-\varphi(r)}{r^{2}} \mathbf{r}^{0}, r\right)=3 \frac{\varphi(r)}{r}+\frac{r \varphi^{\prime}(r)-\varphi(r)}{r}=2 \frac{\varphi(r)}{r}+\varphi^{\prime}(r)
$$

## Problems for Independent Solution

131. For which function $\psi(r)$ will div $\psi(r) r=2 \psi(r)$ ?
132. Find $\operatorname{div}\left(r^{4} r\right)$.
133. Find the divergence of the vector field $2=[c, r]$, where $c$ is a constant vector.
134. Find div $(r|w, r|)$, where $w$ is a constant vector.
135. For which function $\psi(r)$ will the divergence of the field $v=x x l+y j+\psi(r) k$ be independent of $