This post uses index notation, so please consult the linked post for context, if you feel it is necessary, or experience confusion.
Irrotationality
An irrotational vector field is one with a vanishing curl,
\[\begin{equation} \curl \vec{A} = 0 \tag{1} \label{eqn:irroty} \end{equation}\]Irrotational vector fields frequently appear in physical applications, e.g., potential flow, electrostatics, etc.. The reason is that they permit a simplification from needing to solve a linear system of coupled PDEs, to needing to solve only a single linear PDE because one can write $\vec{A} = -\grad \phi$ for a sufficiently smooth scalar function, $\phi$. Inserting this into the above, we would have,
\[\begin{equation} \curl \grad \phi = 0 \tag{2} \label{eqn:curlgradphi} \end{equation}\]which we can take the determinant of in order to verify that Eqn. (\ref{eqn:curlgradphi}) is a valid mathematical statement,
\[\begin{align} \curl \grad \phi &= \begin{vmatrix} \hat{x} & \hat{y} & \hat{z} \\ \partial_{x} & \partial_{y} & \partial_{z} \\ \partial_{x}\phi & \partial_{y}\phi & \partial_{z}\phi \end{vmatrix} \\ &= \hat{x}\left(\partial_{y}\partial_{z}\phi - \partial_{z}\partial_{y}\phi\right) - \hat{y}\left(\partial_{x}\partial_{z}\phi - \partial_{z}\partial_{x}\phi\right) + \hat{z}\left(\partial_{x}\partial_{y}\phi - \partial_{y}\partial_{x}\phi\right) \\ &= 0 \end{align}\]Where we have used the fact that, for a sufficiently smooth $\phi$, $\partial_{i}\partial_{j}\phi = \partial_{j}\partial_{i}\phi$. This was not so demanding, but we can also do the same with index notation, and avoid having to remember how to take th determinant of a 3x3 matrix,
\[\begin{align} \curl \grad \phi &= \hat{e}_{i}\epsilon_{ijk}\partial_{j}\partial_{k}\phi \\ &= \hat{x}\left(\epsilon_{xyz}\partial_{y}\partial_{z}\phi + \epsilon_{xzy}\partial_{z}\partial_{y}\phi\right) + \hat{y}\left(\epsilon_{yzx}\partial_{z}\partial_{x}\phi + \epsilon_{yxz}\partial_{x}\partial_{z}\phi\right) + \hat{z}\left(\epsilon_{zxy}\partial_{x}\partial_{y}\phi + \epsilon_{zyx}\partial_{y}\partial_{x}\phi\right) \\ &= \hat{x}\left(\partial_{y}\partial_{z}\phi - \partial_{z}\partial_{y}\phi\right) + \hat{y}\left(\partial_{z}\partial_{x}\phi - \partial_{x}\partial_{z}\phi\right) + \hat{z}\left(\partial_{x}\partial_{y}\phi - \partial_{y}\partial_{x}\phi\right) \\ &= 0 \end{align}\]