Euclidean Killing Field Question

[Kreizhn]

Hey,

This may seem like a simple question, but hopefully someone can answer it quickly.

Consider the Euclidean 2-metric $ds^2 = dx^2 + dy^2$. There are three killing fields, two translations
$$K_1 = \frac{\partial}{\partial x}, \qquad K_2 = \frac{\partial}{\partial y}$$
and a rotation. Now my issue is this, if $K_3$ is the rotational Killing field, with coordinate decomposition
$$K_3 = K_3^x \frac{\partial}{\partial x} + K_3^y \frac{\partial}{\partial y}$$
We can show that $K_3$'s components must satisfy the differential question
$$\frac{\partial K^y}{\partial x} + \frac{\partial K^x}{\partial y} = 0$$
Hence we can have two solutions $(K^x,K^y) = (-y,x)$ and $(K^x,K^y) = (y, -x)$. Clearly these are just a multiple of -1 different. But if one were to plot these, they would give rotations in opposite directions. One would move clockwise, the other counter-clockwise.

Do we care about the orientation of the field? Or just it's general flow? Do these both represent the same field all the same?

 

[Kreizhn]

Is it because the flow (as a one parameter semi-group) can move in both directions? Hence the flow generated by each is equivalent, and just corresponds to taking a negative "time?"