题目
单选题（共20题，80.0分） 19.（4.0分）函数u=e^xcos(yz)在点(0,0,0)处沿方向vec(l)=(2,1,-2)的方向导数是 A. -(2)/(3) B. (2)/(3) C. (1)/(3) D. -(1)/(3)
单选题（共20题，80.0分） 19.（4.0分）函数$u=e^{x}\cos(yz)$在点(0,0,0)处沿方向$\vec{l}=(2,1,-2)$的方向导数是
A. $-\frac{2}{3}$
B. $\frac{2}{3}$
C. $\frac{1}{3}$
D. $-\frac{1}{3}$
A. $-\frac{2}{3}$
B. $\frac{2}{3}$
C. $\frac{1}{3}$
D. $-\frac{1}{3}$
题目解答
答案
1. **计算梯度**
\[
\nabla u = \left( e^x \cos(yz), -e^x z \sin(yz), -e^x y \sin(yz) \right)
\]
2. **在点(0,0,0)处评估梯度**
\[
\nabla u \bigg|_{(0,0,0)} = (1, 0, 0)
\]
3. **归一化方向向量**
\[
\vec{l} = (2, 1, -2) \quad \Rightarrow \quad $\vec{l}$ = 3 \quad \Rightarrow \quad \hat{l} = \left( \frac{2}{3}, \frac{1}{3}, -\frac{2}{3} \right)
\]
4. **计算方向导数**
\[
D_{\vec{l}} u = \nabla u \cdot \hat{l} = (1, 0, 0) \cdot \left( \frac{2}{3}, \frac{1}{3}, -\frac{2}{3} \right) = \frac{2}{3}
\]
**答案：** $\boxed{B}$