评分及理由

（1）得分及理由（满分4分）

学生作答为"i-k"，这表示向量$\vec{i} - \vec{k}$，与标准答案$\vec{i} - \vec{k}$完全一致。计算旋度$\text{rot}\vec{F}$（即$\nabla \times \vec{F}$）的公式为： $$ \nabla \times \vec{F} = \left( \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} \right)\vec{i} - \left( \frac{\partial F_z}{\partial x} - \frac{\partial F_x}{\partial z} \right)\vec{j} + \left( \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right)\vec{k} $$ 其中$\vec{F} = (xy, -yz, xz)$。计算各分量： - $i$分量：$\frac{\partial (xz)}{\partial y} - \frac{\partial (-yz)}{\partial z} = 0 - (-y) = y$ - $j$分量：$-\left[ \frac{\partial (xz)}{\partial x} - \frac{\partial (xy)}{\partial z} \right] = -[z - 0] = -z$ - $k$分量：$\frac{\partial (-yz)}{\partial x} - \frac{\partial (xy)}{\partial y} = 0 - x = -x$ 因此$\text{rot}\vec{F} = y\vec{i} - z\vec{j} - x\vec{k}$。在点$(1,1,0)$处，代入得$1\cdot\vec{i} - 0\cdot\vec{j} - 1\cdot\vec{k} = \vec{i} - \vec{k}$。学生答案正确且无逻辑错误，得满分4分。

题目总分：4分