The Curl of a Vector as a Sum of Cross Products
\[\mathbf{v}=v_1 \mathbf{i} + v_2 \mathbf{j} + v_2 \mathbf{k}\]
where each of \[v_1, \: v_2 \, \: v_3 \]
are functions of \[x, \; y, \; \\]
is\[ (\frac{\partial}{\partial x} \mathbf{i}+\frac{\partial}{\partial y} \mathbf{j}+ \frac{\partial}{\partial k} \mathbf{k}) \times (v_1 \mathbf{i} + v_2 \mathbf{j} + v_3 \mathbf{k}) = (\frac{\partial v_3}{\partial y} - \frac{\partial v_2}{\partial z}) \mathbf{i} +( \frac{\partial v_1}{\partial z} - \frac{\partial v_3}{\partial x}) \mathbf{i} + (\frac{\partial v_2}{\partial x} - \frac{\partial v_1}{\partial y}) \mathbf{k}\]
We can write this is matrix form as
\[ \left| \begin{array}{ccc} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial }{\partial x} & \frac{\partial }{\partial y} &\frac{\partial }{\partial z} \\ v_1 & v_2 & v_3 \end{array} \right|\]
We can also write
\[- \frac{\partial v_3}{\partial x} \mathbf{j} + \frac{\partial v_2}{\partial x} \mathbf{k} = \left| \begin{array}{ccc} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & 0 & 0 \\ \frac{\partial v_1}{\partial x} & \frac{\partial v_2}{\partial x} & \frac{\partial v_3}{\partial x} \end{array} \right| = \mathbf{i} \times \frac{\partial \mathbf{v}}{\partial x}\]
\[\frac{\partial v_3}{\partial y}\mathbf{i} - \frac{\partial v_1}{\partial y} \mathbf{k} = \left| \begin{array}{ccc} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 0 & 1 & 0 \\ \frac{\partial v_1}{\partial y} & \frac{\partial v_2}{\partial y} & \frac{\partial v_3}{\partial y} \end{array} \right| = \mathbf{j} \times \frac{\partial \mathbf{v}}{\partial y}\]
\[-\frac{\partial v_2}{\partial z} \mathbf{i} + \frac{\partial v_1}{\partial z} \mathbf{j} = \left| \begin{array}{ccc} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 0 & 0 & 1 \\ \frac{\partial v_1}{\partial z} & \frac{\partial v_2}{\partial z} & \frac{\partial v_3}{\partial z} \end{array} \right| = \mathbf{k} \times \frac{\partial \mathbf{v}}{\partial z}\]
\[\mathbf{\nabla} \times \mathbf{v} = \mathbf{i} \times \frac{\partial \mathbf{v}}{\partial x} + \mathbf{j} \times \frac{\partial \mathbf{v}}{\partial y} + \mathbf{k} \times \frac{\partial \mathbf{v}}{\partial z}\]
