I am reading Brocker and tom Dieck's "Representations of Compact Lie groups". In the proof of establishing the degree of a map between 2 oriented (5.19 in Chapter 1), compact, n-dimensional manifolds, they construct a ball B about a point q $\in$ N, the sets $\phi$(B) cover N as $\phi$ runs through diffeomorphisms homotopic to the identity.
In a special case, if x $\in$ N and q are both contained in a compact ball of a chart domain, they claim the $\phi$ can be constructed by integrating an appropriate vector field which vanishes outside the ball. Moreover, the claim is that $\phi$(q) = x.
My question is how do we pick this vector field. Do we use an integral curve connecting x and q and get the vector field using the flow? Thank you.
By rotating/expanding/contracting the chart, you can assume that the chart takes values in the open ball $B(\mathcal O,2) \subset \mathbb R^n$, such that in this chart we have $q = \mathcal O = (0,0,....,0)$ and $x = (1,0,...,0)$.
Choose a smooth function $f : B(\mathcal O,2) \to \mathbb R$ taking values in $[0,1]$ which is equal to $1$ on the line segment $\overline{qx}$, and is equal to $0$ outside of the ball $B(\mathcal O,1.5)$.
The vector field you need on this chart is simply $f(p) \cdot \frac{\partial}{\partial x_1}$, for $p \in B(\mathcal O,2)$. The time 1 map of the flow generated by that vector field takes $q$ to $x$ and is stationary outside of the ball $B(\mathcal O,1.5)$, hence it pastes together smoothly with the flow on $N$ that is stationary outside of the domain of this coordinate chart.