# Euler Lagrange equation and a varying Lagrangian

## Homework Statement:

The variation of the Lagrangian with respect to PSI and X

## Relevant Equations:

Euler Lagrange equation
Hello, I have been working on the three-dimensional topological massive gravity (I'm new to this field) and I already faced the first problem concerning the mathematics, after deriving the lagrangian from the action I had a problem in variating it
Here is the Lagrangian

The first variation should be with respect to Ψ I tried to find this result

but I had a sign different, my result had a Minus sign.

The second variation is with respect to the vector X, note that

and the result is

I tried to use the euler lagrange equation to find this result but i don't have much experience dealing with this wedge product variation