= Solution
Orthogonality and $\|\widetilde\phi(x)\|=\|x\|$ give
$$
\|x+\varepsilon\widetilde\phi(x)\|^2=(1+\varepsilon^2)\|x\|^2,
$$
so $\psi_\varepsilon$ maps each boundary sphere to itself. Also
$$
D\psi_\varepsilon=(1+\varepsilon^2)^{-1/2}
(I+\varepsilon D\widetilde\phi).
$$
On compact $A$, this <derivative> is uniformly close to the identity; for sufficiently small $\varepsilon$ its <determinant> is positive and it is locally invertible. The map is properly homotopic to the identity and has degree one, so the local diffeomorphism has one sheet and is a global diffeomorphism $A\to A$.
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