Lie derivative of a 1-form in the canonical coframe

Let $(x^{1}, \dots, x^{m})$ be local coordinates on a manifold $M$ , with the canonical coframe ${d x^{1}, \dots, d x^{m}}$ and its dual frame ${\partial_{1}, \dots, \partial_{m}}$ . Write

X = X^{j} \partial_{j}, ω = ω_{i} d x^{i} .

Then

(L_{X} ω)_{i} = X^{j} \partial_{j} ω_{i} + ω_{j} \partial_{i} X^{j}

with the Einstein summation convention. This is the coordinate (connection-free) form of the formula for $L_{X}$ on 1-forms.

Derivation from Cartan's formula

By the Cartan formula, $L_{X} ω = d (i_{X} ω) + i_{X} d ω$ . The two terms are

i_{X} ω = X^{j} ω_{j} ⟹ d (i_{X} ω) = \partial_{i} (X^{j} ω_{j}) d x^{i} = (ω_{j} \partial_{i} X^{j} + X^{j} \partial_{i} ω_{j}) d x^{i},

i_{X} d ω = (X^{j} \partial_{j} ω_{i} - X^{j} \partial_{i} ω_{j}) d x^{i},

where in the second line $d ω = \frac{1}{2} (\partial_{j} ω_{k} - \partial_{k} ω_{j}) d x^{j} \land d x^{k}$ and $i_{X} (d x^{j} \land d x^{k}) = X^{j} d x^{k} - X^{k} d x^{j}$ . Adding both contributions, the terms $X^{j} \partial_{i} ω_{j}$ cancel and the formula follows. $◼$