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- How do I know whether or not something is a Killing field?

I'm trying to understand Killing fields and I've hit a bump right away. Suppose I'm in a plane and using the Cartesian coordinate system, and I consider the swirly vector field [itex] \vec{v}=(-y,x) [/itex]. It is easy to show that [itex] \nabla _{y} v^x + \nabla_{x} v^y = -1+1=0 [/itex]. But if I convert this to polar coordinates I get [itex] \vec{v}=(0,1) [/itex] and [itex] \nabla _{\theta} r + \nabla_{r}\theta = -r + 1/r \ne 0 [/itex] . So either I'm computing my covariant derivative components wrong, I'm misusing Killing's equation or (and I doubt this) whether or not a vector field is a Killing field depends on one's choice of coordinates.

Any feedback would be appreciated!

Any feedback would be appreciated!

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