Example 3. Calculate div $(u \mathbf{a})$, where $u(M)$ is a scalar function, $\mathbf{z}(M)=P(x, y, z) \mathbf{i}+Q(x, y, z) \mathbf{j}+R(x, y, z) \mathbf{k}$ is a vector function.
Solution. Using formula (3), we find

$v(u z)=\frac{\partial(u P)}{\partial x}+\frac{\partial(u Q)}{\partial y}+\frac{\partial(u R)}{\partial z}=u \frac{\partial P}{\partial x}+P \frac{\partial u}{\partial x}+u \frac{\partial Q}{\partial y}+Q \frac{\partial u}{\partial y}+$

$$
+u \frac{\partial R}{\partial x}+R \frac{\partial u}{\partial z}=u\left(\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}\right)+P \frac{\partial u}{\partial x}+Q \frac{\partial u}{\partial y}+R \frac{\partial u}{\partial z}=
$$

$=\boldsymbol{u} \cdot \operatorname{div} \mathbf{a}+(\mathbf{a}$, grad $\boldsymbol{u})$

$$
\operatorname{div}(u \mathbf{z})=u \cdot \operatorname{div} \mathbf{a}+(\mathbf{a}, \mathrm{grad} u) . \quad D
$$