Let $f: N \to M$ be a smooth map. Then in Bott and Tu they define a relative de Rham complex $\Omega^{\bullet}(f)$ where $\Omega^k(f) = \Omega^k(M) \oplus \Omega^{k-1}(N)$ with coboundary map given in terms of the ordinary exterior derivative $d$ by $(\omega, \theta) \mapsto (d\omega, f^*\omega - d\theta)$. The cohomology of this complex is denoted $H^q(f)$.
I am wondering if there are any examples (other than $f$ being the identity) where it is possible to compute $H^k(f)$ directly from the definition, without using the accompanying long exact sequence in cohomology? $$\ldots \to H^{k-1}(f) \to H^{k-1}(M) \to H^{k-1}(N) \to H^k(f) \to H^k(M) \to H^k(N) \to \ldots$$
As far as I can tell, Bott and Tu don't have any examples where they do this.
In particular, I am interested in using this for the embedding $i: \partial M \hookrightarrow M$ of the boundary into a manifold with boundary.
I am playing around with a simple toy example, with $M$ being the closed unit ball in $\mathbb{R}^n$, and $\partial M$ being the unit sphere, but unfortunately my algebraic topology background is a bit weak so I can't see how to compute without resorting to this long exact sequence.