Can we get short formula to get the curl of a function $ M \hat{i} + N \hat{j} +O \hat{k} $

Question

Can we get short formula to get the curl of a function $$M \hat{i} + N \hat{j} +O \hat{k} $$. For $$M \hat{i} + N \hat{j}$$ we know $\partial N /\partial x - \partial M/\partial y $ would give the curl. But do we have something for 3 dimensional vector fields also ?

Answer 1

In 3D, $$\text{curl } \hat{F} = \nabla \times \hat{F} = \det \begin{bmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial }{\partial x} & \frac{\partial }{\partial y} & \frac{\partial }{\partial z} \\ M & N & O \end{bmatrix}$$ where $$\hat{F} = M \hat{i} + N \hat{j} + O \hat{k}$$