题目
. 设 overline (F)(x,y,z)= {e)^xsin y,2x(y)^2+z,xz(y)^2} , 则 overline (F)|=(1,0,1)=
题目解答
答案
解析
步骤 1：计算 $\overrightarrow {F}(x,y,z)$ 的散度
散度的定义为：$div\overrightarrow {F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}$，其中 $\overrightarrow {F} = \{P, Q, R\}$。
步骤 2：计算 $\frac{\partial P}{\partial x}$, $\frac{\partial Q}{\partial y}$, $\frac{\partial R}{\partial z}$
- $\frac{\partial P}{\partial x} = \frac{\partial}{\partial x}({e}^{x}\sin y) = {e}^{x}\sin y$
- $\frac{\partial Q}{\partial y} = \frac{\partial}{\partial y}(2x{y}^{2}+z) = 4xy$
- $\frac{\partial R}{\partial z} = \frac{\partial}{\partial z}(xz{y}^{2}) = xy^2$
步骤 3：将上述结果代入散度公式
$div\overrightarrow {F} = {e}^{x}\sin y + 4xy + xy^2$
步骤 4：将点 $(1,0,1)$ 代入散度公式
$div\overrightarrow {F}|(1,0,1) = {e}^{1}\sin 0 + 4(1)(0) + (1)(0)^2 = 0$
散度的定义为：$div\overrightarrow {F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}$，其中 $\overrightarrow {F} = \{P, Q, R\}$。
步骤 2：计算 $\frac{\partial P}{\partial x}$, $\frac{\partial Q}{\partial y}$, $\frac{\partial R}{\partial z}$
- $\frac{\partial P}{\partial x} = \frac{\partial}{\partial x}({e}^{x}\sin y) = {e}^{x}\sin y$
- $\frac{\partial Q}{\partial y} = \frac{\partial}{\partial y}(2x{y}^{2}+z) = 4xy$
- $\frac{\partial R}{\partial z} = \frac{\partial}{\partial z}(xz{y}^{2}) = xy^2$
步骤 3：将上述结果代入散度公式
$div\overrightarrow {F} = {e}^{x}\sin y + 4xy + xy^2$
步骤 4：将点 $(1,0,1)$ 代入散度公式
$div\overrightarrow {F}|(1,0,1) = {e}^{1}\sin 0 + 4(1)(0) + (1)(0)^2 = 0$