Whats $\epsilon_{ijk} $ in Vector Calculus?

Question

I've seen $\epsilon_{ijk} $ used to prove properties of the $\nabla $ operators on both scalar functions and vector fields, but I don't understand what it is. Can somebody explain what this is?

Answer 1

It is the Levi-Civita symbol in three dimension

$$ \varepsilon_{ijk} = \begin{cases} +1 & \text{if } (i,j,k) \text{ is } (1,2,3), (2,3,1), \text{ or } (3,1,2), \\ -1 & \text{if } (i,j,k) \text{ is } (3,2,1), (1,3,2), \text{ or } (2,1,3), \\ \;\;\,0 & \text{if } i = j, \text{ or } j = k, \text{ or } k = i \end{cases}$$