Problem
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 $i_{X} \mathrm{~d} \alpha=0$ ?