I am reading Milnor & Stasheff's Characteristic classes (page 98 to be precise). After defining the Euler class, they state two basic properties claiming that the proofs are obvious:
1.(Naturality). If $f: B \to B'$ is covered by an orientation preserving bundle map $\xi\to\xi'$, then $e(\xi)=f^*e(\xi')$
2.If the orientation of $\xi$ is reversed, then the Euler class $e(\xi)$ changes sign.
I think for the first one we need to check the definition for $e(\xi)$ is staistied for $f^*e(\xi')$. But I'm not sure how to proceed. In the second I don't know how to reverse orientation for a bundle.
Kindly Help!!
Any kind of Hint will be really helpful.
Regards