问题
已知 $\boldsymbol{A}(x, y, z)$ $=$ $P(x, y, z)$ $\boldsymbol{i}$ $+$ $Q(x, y, z)$ $\boldsymbol{j}$ $+$ $R(x, y, z)$ $\boldsymbol{k}$, 则旋度 $\boldsymbol{r o t} \mathbf{A}$ $=$ $?$
选项
[A]. $\boldsymbol{r o t} \mathbf{A}$ $=$ $($ $\frac{\partial R}{\partial y}$ $+$ $\frac{\partial Q}{\partial z}$ $)$ $\boldsymbol{i}$ $-$ $($ $\frac{\partial P}{\partial z}$ $+$ $\frac{\partial R}{\partial x}$ $)$ $\boldsymbol{j}$ $-$ $($ $\frac{\partial Q}{\partial x}$ $+$ $\frac{\partial P}{\partial y}$ $)$ $\boldsymbol{k}$ $=$ $\left|\begin{array}{ccc} \boldsymbol{i} & \boldsymbol{j} & \boldsymbol{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ P & Q & R \end{array}\right|$[B]. $\boldsymbol{r o t} \mathbf{A}$ $=$ $($ $\frac{\partial R}{\partial y}$ $-$ $\frac{\partial Q}{\partial z}$ $)$ $\boldsymbol{i}$ $+$ $($ $\frac{\partial P}{\partial z}$ $-$ $\frac{\partial R}{\partial x}$ $)$ $\boldsymbol{j}$ $+$ $($ $\frac{\partial Q}{\partial x}$ $-$ $\frac{\partial P}{\partial y}$ $)$ $\boldsymbol{k}$ $=$ $\left|\begin{array}{ccc} \boldsymbol{i} & \boldsymbol{j} & \boldsymbol{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ P & Q & R \end{array}\right|$[C]. $\boldsymbol{r o t} \mathbf{A}$ $=$ $($ $\frac{\partial R}{\partial y}$ $-$ $\frac{\partial Q}{\partial z}$ $)$ $\boldsymbol{i}$ $+$ $($ $\frac{\partial P}{\partial z}$ $-$ $\frac{\partial R}{\partial x}$ $)$ $\boldsymbol{j}$ $+$ $($ $\frac{\partial Q}{\partial x}$ $-$ $\frac{\partial P}{\partial y}$ $)$ $\boldsymbol{k}$ $=$ $\left|\begin{array}{ccc} P & Q & R \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ \boldsymbol{i} & \boldsymbol{j} & \boldsymbol{k} \end{array}\right|$[D]. $\boldsymbol{r o t} \mathbf{A}$ $=$ $($ $\frac{\partial R}{\partial y}$ $-$ $\frac{\partial Q}{\partial z}$ $)$ $\boldsymbol{i}$ $\times$ $($ $\frac{\partial P}{\partial z}$ $-$ $\frac{\partial R}{\partial x}$ $)$ $\boldsymbol{j}$ $\times$ $($ $\frac{\partial Q}{\partial x}$ $-$ $\frac{\partial P}{\partial y}$ $)$ $\boldsymbol{k}$ $=$ $\left|\begin{array}{ccc} \boldsymbol{i} & \boldsymbol{j} & \boldsymbol{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ P & Q & R \end{array}\right|$
答 案 ![下一题 - 荒原之梦](/wp-content/plugins/CoreEngine/image/button_right_green.svg)
$\boldsymbol{r o t} \mathbf{A}$ $=$ $($ $\frac{\partial R}{\partial y}$ $-$ $\frac{\partial Q}{\partial z}$ $)$ $\boldsymbol{i}$ $+$ $($ $\frac{\partial P}{\partial z}$ $-$ $\frac{\partial R}{\partial x}$ $)$ $\boldsymbol{j}$ $+$ $($ $\frac{\partial Q}{\partial x}$ $-$ $\frac{\partial P}{\partial y}$ $)$ $\boldsymbol{k}$ $=$ $\left|\begin{array}{ccc} \boldsymbol{i} & \boldsymbol{j} & \boldsymbol{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ P & Q & R \end{array}\right|$