I always get confused with the Lie derivative stuff so sorry if this is a stupid question.
Consider a loop $u(z) : S^1 \rightarrow M$ in manifold $M$. For a variantion of $u$ along direction $X$, denote as $u'$, i.e. $u'(t,z): (-\epsilon,\epsilon)\times S^1 \rightarrow M$ that satisfies $u'(0,z)=u(z)$ and $\partial_tu'(0,z)=X$ where $X$ is a vector field along $u(z)$.
Consider an arbitary form $T$, how do we see that $\frac{d}{dt}u'^*T|_{t=0}=u^*(\mathcal{L}_XT)$.
I tried to write down the definition for Lie derivative but still don't know how to do this.