= Solution
Write the <covector field> as $\omega=\omega_\mu dx^\mu$. From the contraction and <product rule> properties,
$$
(\mathcal L_X\omega)(Y)=X(\omega(Y))-\omega([X,Y]).
$$
Take $Y=\partial_\mu$. The <Lie bracket of vector fields> is $[X,\partial_\mu]=-(\partial_\mu X^\nu)\partial_\nu$, so
$$
\boxed{(\mathcal L_X\omega)_\mu
=X^\nu\partial_\nu\omega_\mu+\omega_\nu\partial_\mu X^\nu.}
$$
Equivalently, the covariant-factor contribution comes from $\mathcal L_Xdx^\mu=dX^\mu=(\partial_\nu X^\mu)dx^\nu$. The formula holds in any <coordinate basis> and is the one-form case of the <coordinate tensor Lie derivative>.
Back to article page
