Question involving Levi-Civita symbol

[sunnyskies]

Can someone please explain to me why

\epsilon_{ijk}\frac{\partial}{\partial x_i}\frac{\partial A_k}{\partial x_j} = 0

where A is a constant vector field.

 

[kevinferreira]

The double differentiation w.r.t the coordinates is symmetric: $$\frac{\partial}{\partial x_j}\frac{\partial}{\partial x_i}=\frac{\partial}{\partial x_i}\frac{\partial}{\partial x_j}$$

On the other hand, the Levi-Civita tensor is anti-symmetric on i,j. Interchange these indices and you therefore get: $$\epsilon_{ijk}\frac{\partial}{\partial x_i}\frac{\partial}{\partial x_j}=-\epsilon_{jik}\frac{\partial}{\partial x_j}\frac{\partial}{\partial x_i}=-\epsilon_{ijk}\frac{\partial}{\partial x_i}\frac{\partial}{\partial x_j}$$
where in the second equality we have renamed the dummy indices (they are summed over) i to j and j to i.
So we get that something is equal to minus itself, and thus is zero.

 

[sunnyskies]

Makes perfect sense, thank you!