Find curl \(\mathbf{F}\) if \(\mathbf{F}=x^{2}y\mathbf{i}-2xz\,\mathbf{j} +2yz\,\mathbf{k}\).

Solution  \[ \begin{eqnarray*} {\rm curl}\,\mathbf{F}& =&\left\vert \begin{array}{@{}c@{\quad}c@{\quad}c} \mathbf{i} & \mathbf{j} & \mathbf{k}\\ \dfrac{\partial }{\partial x} & \dfrac{\partial }{\partial y} & \dfrac{ \partial }{\partial z}\\ x^{2}y & -2xz & 2yz \end{array} \right\vert \\ & =&\left[ \dfrac{\partial }{\partial y}(2yz)-\dfrac{\partial }{\partial z} (-2xz)\right] \,\mathbf{i}-\left[ \dfrac{\partial }{\partial x}(2yz)-\dfrac{ \partial }{\partial z}(x^{2}y)\right] \,\mathbf{j}+\left[ \dfrac{\partial }{ \partial x}(-2xz)-\dfrac{\partial }{\partial y}(x^{2}y)\right] \mathbf{k} \\ & =&(2z+2x)\,\mathbf{i}-(0-0)\,\mathbf{j}+(-2z-x^{2})\mathbf{k}=(2z+2x)\, \mathbf{i}+(-2z-x^{2})\,\mathbf{k} \end{eqnarray*} \]