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## Homework Statement

Find all Killing vector solutions of the metric

[tex]g_x{_x}=x^2, g_x{_y}=g_y{_x}=0, g_y{_y}=x[/tex]

where [itex](x^a)=(x^0, x^1)=(x, y)[/itex]

## Homework Equations

Killing equations:

[tex]L_Xg_a{_b} = X^e\partial_eg_a{_b}+g_a{_d}\partial_bX^d+g_b{_d}{\partial}_aX^d = 0[/tex]

## The Attempt at a Solution

[tex]L_Xg_x{_x} = X^a+x\partial_xX^a=0[/tex]

[tex]L_Xg_x{_y}= L_Xg_y{_x}=x\partial_yX^a+\partial_xX^a=0[/tex]

[tex]L_Xg_y{_y}=X^a+2x\partial_yX^a=0[/tex]

In the back of the book, it says the solution is [itex]\frac{\partial}{{\partial}y}[/itex]. I don't really know what they mean by that. I've always seen [itex]\frac{\partial}{{\partial}y}[/itex] as an operation that takes the partial derivative of something with respect to y, not a value. I thought perhaps it meant any function that depends only on y and not on x, but if I plug that into the equations above, I get:

[tex]f(y)\neq0[/tex]

[tex]x\partial_yf\neq0[/tex]

[tex]f(y)+2x\partial_yf\neq0[/tex]

I'm pretty sure I've got the equations correct; I just don't know what they mean when they say the solution is [itex]\frac{\partial}{{\partial}y}[/itex].