Let $U,V$ be connected open subsets of $\Bbb R^n$ and let $f\colon U \to V$ be a $C^{\infty}$ proper map. For all $w \in \Omega_c^n(V)$,
$\int_{U}f^*(w)=\gamma \int_Vw$.
Now define $\deg(f)=\gamma$, where $\gamma$ is defined as above.
a) Let $f\colon \Bbb R^n \to \Bbb R^n$ be the translation, $f(x)=x+a$. Show that $\deg(f)=1$ (Given hint is show that this is true for $n=1$ case. For arbitrary case $w=\phi(x) dx_1 dx_2\cdots dx_n$, where $\phi(x)=\psi(x_1) \cdots \psi(x_n)$ and $\psi\colon \Bbb R \to \Bbb R$ such that $\int \psi(t)dt=\int \psi(t-a)dt$.)
b) Let $\sigma \in S_n$ be a permutation of numbers and let $f_{\sigma}(x_1,...,x_n)=(x_{\sigma(1)},...,x_{\sigma(n)})$. Compute $\deg(f_{\sigma})$. I think taking same $w$ as above will solve my purpose by computing $f^*_{\sigma}(w)$.
c) Compute degree of any linear map $f\colon\Bbb R^n \to \Bbb R^n$.
How to deal with these kind of problems? Any idea? If anyone can give glimps of any one of these please post.