killing vector help

jiggers

noob here
* indicates multiply (or 'operate on'), d_c is partial derivative w.r.t. c

tensor indices have always troubled me, my problem this time is im trying to prove a vector E = (-y*d_x +x*d_y) is a killing vector after having computed the connection coefficients for 2-d riemannian manifold, diagonal metric from ds^2 = f(x,y)*(dx^2 + dy^2)

Im looking at the Lie derivative statement of Killing's eqns,
E^c*d_c*g_ab - g_ac*d_c*E^b - g^cb*d_c*E^a

how do i do that explicitly, showing each term for x and y? how can there be terms involving a,b and c, when i only have x and y to work with?
any links to examples of "show this is a killing vector" would be a great help.