I'm trying to do some simple calculations with differential forms, and I'd appreciate some help. For example, Why is the following wrong?
Consider $L_{\sin x\partial_y}(\sin x dy)$ in $\mathbb{R}^3$.
In general, for $f,g\in C^\infty(M)$, $X\in\Gamma(TM)$ and $\omega\in\Omega^\bullet(M)$ we have - $$L_{fX}\omega=fL_X\omega+df\wedge(i_X\omega)$$ and - $$L_X(f\omega)=(i_Xdf)\omega+fL_X\omega$$ combining the two, we get - $$L_{fX}(g\omega)=fL_X(g\omega)+df\wedge\big(i_X(g\omega)\big)=(fi_Xdg)\omega+fg(L_X\omega)+df\wedge\big(i_X(g\omega)\big)$$ Thus - $$ \begin{aligned}[t] L_{\sin x\partial_y}(\sin x dy)&= (\sin xi_{\partial_y}d\sin x)dy+\sin^2x(L_{\partial_y}dy)+d(\sin x)\wedge\big(i_{\partial_y}(\sin xdy)\big)\\&= \sin^2x+\sin xd(\sin x)=\sin x(\sin x+\cos x dx) \end{aligned} $$
To my surprise, while there's no shortage of resources about analysis on manifolds, I'm having trouble finding some neat "cheat-sheets" of formulas and identities to work with, or even just examples of simple calculations. Does someone have a suggestion for a good concise resource?
Just use Cartan's magic formula: $$\begin{align}\mathcal{L}_{\sin x\,\partial_y} (\sin x\,{\rm d}y) &= \iota_{\sin x\,\partial_y}({\rm d}(\sin x\,{\rm d}y)) + {\rm d}(\iota_{\sin x\,\partial_y}(\sin x\,{\rm d}y)) \\ &= \iota _{\sin x\,\partial_y}(\cos x\,{\rm d}x\wedge {\rm d}y) + {\rm d}(\sin^2x) \\ &=\cos x\begin{vmatrix}0 & {\rm d}x \\ \sin x & {\rm d}y\end{vmatrix} +2\sin x\cos x\,{\rm d}x \\ &= - \sin x \cos x\,{\rm d}x +2\sin x\cos x\,{\rm d}x \\ &= \sin x\cos x\,{\rm d}x .\end{align}$$ I'm not sure what happened in your calculation. It looked like you wrote $\mathcal{L}_{\partial_y}({\rm d}y)=1$, but this should be a $1$-form and not a function. Since $\mathcal{L}_X$ is a derivation, you would have that $$\mathcal{L}_X({\rm d}y)(Y) = X({\rm d}y (Y)) - {\rm d}y([X,Y]).$$I reccomend the Analysis and Algebra on Differentiable Manifolds workbook. There's a lot of computations we normally don't see carried out in the classical references.