评分及理由

（1）得分及理由（满分4分）

学生作答结果为：\(\overrightarrow{i}-\overrightarrow{k}\)。该题要求计算向量场\(\boldsymbol{F}(x,y,z) = xy\boldsymbol{i} - yz\boldsymbol{j} + zx\boldsymbol{k}\)在点\((1,1,0)\)处的旋度\(\text{rot}\ \boldsymbol{F}\)。

旋度的计算公式为： \[ \text{rot}\ \boldsymbol{F} = \nabla \times \boldsymbol{F} = \begin{vmatrix} \boldsymbol{i} & \boldsymbol{j} & \boldsymbol{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ P & Q & R \end{vmatrix} \] 其中\(\boldsymbol{F} = P\boldsymbol{i} + Q\boldsymbol{j} + R\boldsymbol{k}\)，即\(P = xy, Q = -yz, R = zx\)。

计算得： \[ \text{rot}\ \boldsymbol{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right)\boldsymbol{i} - \left( \frac{\partial R}{\partial x} - \frac{\partial P}{\partial z} \right)\boldsymbol{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right)\boldsymbol{k} \] 代入函数： \[ \frac{\partial R}{\partial y} = \frac{\partial (zx)}{\partial y} = 0, \qua...