Задача 44. Вычислить $\mathrm{\,grad\,} \vec{A}(r) \cdot \vec{r}$, $\mathrm{\,grad\,} \vec{A}(r) \cdot \vec{B}(r)$, $\mathrm{\,div\,} \varphi(r) \vec{A}(r)$, $\mathrm{\,rot\,} \varphi(r) \vec{A}(r)$, $(\vec{l}\cdot\nabla) \varphi(r) \vec{A}(r)$.

Решение.

Найдём:

\[\mathrm{\,div\,} \vec{A}(r) = \frac{\partial A_x}{\partial r} \frac{\partial r}{\partial x} + \ldots = \frac{\partial A_x}{\partial r} \frac{x}{r} + \ldots = \frac{\vec{A}'\cdot\vec{r}}{r}\] \[\mathrm{\,rot\,} \vec{A}(r) = \begin{vmatrix} \vec{e}_x & \vec{e}_y & \vec{e}_z \\ \dfrac{\partial}{\partial x} & \dfrac{\partial}{\partial y} & \dfrac{\partial}{\partial z} \\ A_x(r) & A_y(r) & A_z(r) \end{vmatrix} = \begin{vmatrix} \vec{e}_x & \vec{e}_y & \vec{e}_z \\ \dfrac{\partial r}{\partial x} & \dfrac{\partial r}{\partial y} & \dfrac{\partial r}{\partial z} \\ A_x'(r) & A_y'(r) & A_z'(r) \end{vmatrix} = \frac{\vec{r}\times\vec{A}'}{r}\] \[(\vec{l}\cdot\nabla) \vec{A}(r) = \left(l_x \frac{\partial}{\partial x} + l_y \frac{\partial}{\partial y} + l_z \frac{\partial}{\partial z}\right) \vec{A} = \left(l_x \frac{\partial r}{\partial x} + l_y \frac{\partial r}{\partial y} + l_z \frac{\partial r}{\partial z}\right) \vec{A}'= \frac{\vec{l}\cdot\vec{r}}{r} \vec{A}'\]

Теперь можем решить задачу:

\[\mathrm{\,grad\,} \vec{A}(r) \cdot \vec{r} = \vec{A}\times \mathrm{\,rot\,} \vec{r} + \vec{r} \times \mathrm{\,rot\,} \vec{A} + (\vec{r}\cdot\nabla)\vec{A} + (\vec{A}\cdot\nabla)\vec{r} =\] \[= \frac{\vec{r}\times(\vec{r}\times\vec{A}')}{r} + r \vec{A}' + \vec{A} = \frac{\vec{r}(\vec{r}\cdot\vec{A}')}{r} + \vec{A}\] \[\mathrm{\,grad\,} \vec{A}(r) \cdot \vec{B}(r) = \vec{A}\times \mathrm{\,rot\,} \vec{B} + \vec{B} \times \mathrm{\,rot\,} \vec{A} + (\vec{B}\cdot\nabla)\vec{A} + (\vec{A}\cdot\nabla)\vec{B} =\] \[= \frac{\vec{A}\times(\vec{r}\times\vec{B}') + \vec{B}\times(\vec{r}\times\vec{A}') + (\vec{B}\cdot\vec{r})\vec{A}' + (\vec{A}\cdot\vec{r})\vec{B}'}{r} =\] \[= \frac{\vec{r}(\vec{A}\cdot\vec{B})'}{r}\] \[\mathrm{\,div\,} \varphi(r) \vec{A}(r) = \vec{A}\cdot\nabla\varphi + \varphi \mathrm{\,div\,} \vec{A} = \frac{\vec{A}\cdot\vec{r} \varphi' + \varphi \vec{A}' \cdot \vec{r}}{r} = \frac{(\varphi\vec{A})' \cdot \vec{r}}{r}\] \[\mathrm{\,rot\,} \varphi(r) \vec{A}(r) = \nabla\varphi\times\vec{A} + \varphi\nabla\times\vec{A} = \frac{(\varphi \vec{A})'\times \vec{r}}{r}\] \[(\vec{l}\cdot\nabla) \varphi(r) \vec{A}(r) = \frac{\vec{l}\cdot\vec{r}}{r} (\varphi\vec{A})'\]