一、单选题（共20题，80.0分）5.（单选题，4.0分）已知矢量vec(A)=vec(e)_(x)x+vec(e)_(y)x^2+vec(e)_(z)y^2z，计算nablacdotvec(A)=A vec(e)_(x)(1+y^2)B 1+y^2C vec(e)_(x)2yz+vec(e)_(z)2xD 2yz+2x

根据散度公式： \[ \nabla \cdot \vec{A} = \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} + \frac{\partial A_z}{\partial z} \] 将 $\vec{A} = \vec{e}_x x + \vec{e}_y x^2 + \vec{e}_z y^2 z$ 的分量代入： \[ A_x = x, \quad A_y = x^2, \quad A_z = y^2 z \] 计算各偏导数： \[ \frac{\partial A_x}{\partial x} = \frac{\partial}{\partial x}(x) = 1 \] \[ \frac{\partial A_y}{\partial y} = \frac{\partial}{\partial y}(x^2) = 0 \] \[ \frac{\partial A_z}{\partial z} = \frac{\partial}{\partial z}(y^2 z) = y^2 \] 将结果相加： \[ \nabla \cdot \vec{A} = 1 + 0 + y^2 = 1 + y^2 \] 因此，正确答案是 $1 + y^2$。 答案：B. $1 + y^2$