# Commutator-like notation, index notation

## Homework Statement

There are some equations in the notes on field theory I am reading with notation I have never come across before. Someone told me it was a way of ensuring that the expression was anti-symmetric. I can't find it used the same anywhere else but no explanation is provided in the notes which makes me think maybe it is just an ordinary commutator but I am not sure.

## Homework Equations

The equations are...

i. $$\Phi_{\mu\nu} = 2x_{[\mu}\partial_{\nu]}\phi$$
ii. $$\Lambda^{\rho}_{\mu\nu} = 2x_{[\mu}\delta^{\rho}_{\nu]}L$$
iii. $$\partial_{[\mu}F_{\nu\rho]} = 0$$

F is the electromagnetic tensor if it helps.

## The Attempt at a Solution

does the first mean...

$$\left(2x_\mu\partial_\nu\phi - 2x_\nu\partial_\mu\phi\right)$$

but then what do the others mean? For the second I was thinking of ignoring the superscript index - the rho - and doing the same as with the first but then what about the third one or am I not even close?

Thanks

so $2x_{[\mu} \partial_{\nu]} \phi=2 \frac{1}{2!} \left(x_{\mu} \partial_{\nu} - x_{\nu} \partial_{\mu} \right) \phi = \left(x_{\mu} \partial_{\nu} - x_{\nu} \partial_{\mu} \right) \phi=$