VC3.15

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i). \nabla F=\mathrm{curl}F=\begin{vmatrix} i & j & k \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ x^2y & -2xz & 2yz \end{vmatrix}\,

=i[\frac{\partial}{\partial x}(2yz)-\frac{\partial}{\partial z}(-2xz)]-j[\frac{\partial}{\partial}(2yz)-\frac{\partial}{\partial z}(x^2y)]+k[\frac{\partial}{\partial x}(-2xz)-\frac{\partial}{\partial y}(x^2y)]\,

=(2z+2x)i-(0-0)j+(-2z-x^2)k=2(x+z)i-(2z+x^2)k\,

ii). Similarly as in the above,

\nabla F=\mathrm{curl}F=\begin{vmatrix} i & j & k \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ (x^2-y^2) & 2xy & (y^2-2xy) \end{vmatrix}\,

=i[\frac{\partial}{\partial y}(y^2-2xy)-\frac{\partial}{\partial z}(2xy)]-j[\frac{\partial}{\partial x}(y^2-2xy)-\frac{\partial}{\partial z}(x^2-y^2)]+k[\frac{\partial}{\partial x}(2xy)-\frac{\partial}{\partial y}(x^2-y^2)]\,

=(2y-2x-0)i-(-2y-0)j+(2y+2y)k=2(y-x)i+2yj+4yk\,

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