Tensor Fields & Differential Forms Part 2

Problem 2.42 Let $X$ and $Y$ be vector fields on a $C^{\infty}$ manifold $M$. Prove that if $\varphi_{t}$ is the local 1-parameter group generated by $X$, we have for all $p \in M$ : $$ \varphi_{s *}\left(\left(L_{X} Y\right)_{\varphi_{s}^{-1}(p)}\right)=\lim _{t \rightarrow 0} \frac{1}{t}\left(\varphi_{s *} Y_{\varphi_{s}^{-1}(p)}-\varphi_{s+t *} Y_{\varphi_{s+t}^{-1}(p)}\right) $$

Problem 2.43 Let $f$ denote a diffeomorphism of the $C^{\infty}$ manifold $M$. Prove that $$ i_{X}\left(f^{*} \alpha\right)=f^{*}\left(i_{f \cdot X} \alpha\right), \quad X \in \mathfrak{X}(M), \alpha \in \Lambda^{*} M . $$

Problem 2.44 Consider on an open subset of $\mathbb{R}^{3}$ the differential 1-form $$ \alpha=P_{1}(x) \mathrm{d} x^{1}+P_{2}(x) \mathrm{d} x^{2}+P_{3}(x) \mathrm{d} x^{3}, $$ where $x=\left(x^{1}, x^{2}, x^{3}\right)$. (i) Find the conditions under which $i_{X} \mathrm{~d} \alpha=0$ for $$ X=X_{1} \partial / \partial x+X_{2} \partial / \partial y+X_{3} \partial / \partial z $$ (ii) When do we have $i_{X} \alpha=0$ and $i_{X} \mathrm{~d} \alpha=0$ ?