Let $C^k$ action of a compact Lie group $G$ on $R^m$, $D(g)$ denote the differential of the map $x\in R^m \mapsto g(x) \in R^m$ at the origin and $\mu$ is the normalized Haar mesure on $G$, consider a map:
$F(x)=\int_{G} {D(g)^{-1}\left(g(x)\right)}d\mu$
How to calculate the differential $DF$ of $F$ at origin?
Is this right?
$DF(x)=\int_{G} {D(g)^{-1}\left(D(g)(x)\right)}d\mu=\int_{G} {D(g^{-1})D(g)(x)}d\mu=\int_{G} {D(g^{-1}g)(x)}d\mu=x$
Thank you