# Taylor Expansion of tensor moved by a flow.

I am reading Peter Petersen's notes on manifold theory and he introduces Lie Derivatives in the following way.

"Let $X$ be a vector field and $F^t$ the corresponding locally defined flow on a smooth manifold $M$. Thus $F^t (p)$ is defined for small $t$ and the curve $t \to F^t (p)$ is the integral curve for X that goes through $p$ at $t = 0$. The Lie derivative of a tensor in the direction of $X$ is defined as the first order term in a suitable Taylor expansion of the tensor when it is moved by the flow of $X$."

I am uncertain of what he means when he refers to the "Taylor expansion of the tensor as it is moved by the flow of $X$". I am only aware of Taylor expansions for functions and I was hoping that somebody can help clarify this.

Given a smoothly varying $1$-parameter family $\Psi[t]$ of tensor fields, which we can regard as a map $\Psi: J \to \Gamma(\bigotimes^l TM \otimes \bigotimes^k T^*M)$ for some interval $J \ni 0$, we can differentiate $\Psi$ with respect to $t$ and evaluate at $t = 0$ to produce another such family, $$\partial_t \Psi[t] = \lim_{h \to 0} \frac{\Psi[t + h] - \Psi[t]}{h} \in \Gamma({\textstyle \bigotimes^l TM \otimes \bigotimes^k T^*M}).$$ (With respect to any local coordinates, $\Psi[t]$ has some components $\hat{\Psi}[t]^{b_1 \cdots b_l}{}_{a_1 \cdots a_k}$, and the components of $\partial_t \Psi$ are just the usual single-variable derivatives of these with respect to $t$, i.e., $$\widehat{\partial_t \Psi[t]}^{b_1 \cdots b_l}{}_{a_1 \cdots a_k} = \partial_t \hat{\Psi}[t]^{b_1 \cdots b_l}{}_{a_1 \cdots a_k}. )$$
Now, we can just as easily compute higher derivatives $\partial_t^k \Psi[t]$. Then, just as in the familiar case of Taylor series of functions, we can evaluate all of these derivatives at $t = 0$ and assemble the result into a (formal) Taylor series $$\sum_{k = 0}^{\infty} \frac{1}{k!} \partial^k \Psi[t] \vert_{t = 0}.$$ The (coefficient of) the first-order term here is the tensor field $$\phantom{(\ast)} \qquad \partial_t \Psi[t]\vert_{t = 0} = \left.\lim_{h \to 0} \frac{\Psi[h] - \Psi[0]}{h}\right\vert_{t = 0}. \qquad (\ast)$$
Now, given a tensor field $\Phi \in \Gamma(\bigotimes^l TM \otimes \bigotimes^k T^*M)$ and a vector field $X \in \Gamma(TM)$, the flow $\theta_t$ of $X$ gives us (up to the usual issues involved in existence of flows) a $1$-parameter family of tensor fields $$\Phi[t] := \theta_t^* \Phi$$ with $\Phi[0] = \Phi$. Then, substituting $\Phi[t]$ into the formula $(\ast)$ for the first-order coefficient of the series recovers the usual definition of Lie derivative of a tensor field as claim.