= Euclidean Killing vector field
{c}
{title2=$K(x)=A+Bx,\quad B^T=-B$}
On <Euclidean space>, every <Killing vector field> is a constant <translation> generator plus a constant infinitesimal <rotation> generator:
$$
K(\mathbf x)=A+B\mathbf x,\qquad B^T=-B.
$$
Here $A$ is a constant vector and $B$ is an <antisymmetric matrix>. The <second covariant derivative of a Killing vector> vanishes for the <Euclidean metric>, so the components are affine functions of <Cartesian coordinates>; the <Killing equation> makes the linear coefficient antisymmetric.
In the <Euclidean plane> with <metric tensor> $g=dx^2+dy^2$, two useful examples are
$$
K_{\mathrm{tr}}=\partial_x,\qquad
K_{\mathrm{rot}}=-y\partial_x+x\partial_y.
$$
The notation $\partial_x$ denotes a <coordinate basis> vector; $dx$ denotes a <differential one-form>. The corresponding <flow of a vector field> maps are
$$
\Phi_s^{\mathrm{tr}}(x,y)=(x+s,y),\qquad
\Phi_s^{\mathrm{rot}}(x,y)=(x\cos s-y\sin s,\ x\sin s+y\cos s).
$$
For <translation>, all arrows have the same direction and length. For <rotation>, arrows are tangent to <circles>, have length $\sqrt{x^2+y^2}$, and vanish at the origin. The <rotation> flow keeps $x^2+y^2$ fixed, and its nonstationary <integral curves of a vector field> are <circles>. Both flows preserve the <Euclidean metric>. See https://arxiv.org/pdf/gr-qc/9712019#page=148[Carroll's original notes, section 5, equations (5.45)-(5.46)].
\Image[/killing-vector-translation-and-rotation.png]
{title=Euclidean Killing vector fields: translation arrows and straight flow orbits, rotation arrows and circular flow orbits}
The circular <integral curves of a vector field> are symmetry orbits, whereas <geodesics> of the <Euclidean plane> are straight <lines>. A <Killing vector field> does not require its own orbits to be <geodesics>. The <geodesic conserved quantity from a Killing vector> pairs the field with the <velocity> of a separate <geodesic>; it does not require the <velocity> to equal the <Killing vector field>.
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