已知 =x2x+hat (y)xy+z(z)^2 =xy+hat (y)(x)^2,在点 P(2,1,2) 处, 试求:=x2x+hat (y)xy+z(z)^2 =xy+hat (y)(x)^2,

步骤 1：计算 $\overline {A}$ 的散度
对于向量场 $\overline {A}=\hat {x}2x+yxy+z{z}^{2}$，其散度为
$$
D\cdot \overline {A}=\dfrac {\partial }{\partial x}(2x)+\dfrac {\partial }{\partial y}(xy)+\dfrac {\partial }{\partial z}({z}^{2})
$$
步骤 2：计算 $\overline {B}$ 的散度
对于向量场 $\overline {B}=xy+y{x}^{2}$，其散度为
$$
D\cdot \overline {B}=\dfrac {\partial }{\partial x}(y)+\dfrac {\partial }{\partial y}({x}^{2})
$$
步骤 3：计算 $\overline {A}\times \overline {B}$ 的散度
对于向量场 $\overline {A}\times \overline {B}$，其散度为
$$
D\cdot (\overline {A}\times \overline {B})=D\cdot (\overline {A})\times \overline {B}-\overline {A}\times (D\cdot \overline {B})
$$