= Solution
A <local orientation of a manifold> at $x\in M$ is a generator of
$$
H_d(M,M\setminus\{x\};R)\cong R.
$$
An $R$-orientation is a locally coherent choice of such generators. An <R-fundamental class> is a class $[M]_R\in H_d(M;R)$ whose image in every one of these local homology groups is a generator. Those images vary coherently under the restriction maps between small coordinate balls, so an $R$-fundamental class determines an $R$-orientation.
For a closed $R$-oriented $d$-manifold, <Poincare duality> says that cap product with its fundamental class is an isomorphism
$$
-\frown[M]_R:H^k(M;R)\xrightarrow{\sim}H_{d-k}(M;R)
$$
for every $k$.
Choose a generator $[S^d]\in H_d(S^d;\mathbb Z)\cong\mathbb Z$ from the <homology of a sphere>. For every $x\in S^d$, the long exact sequence of $(S^d,S^d\setminus\{x\})$, together with the contractibility of $S^d\setminus\{x\}\cong\mathbb R^d$, shows that
$$
H_d(S^d;\mathbb Z)\xrightarrow{\sim}H_d(S^d,S^d\setminus\{x\};\mathbb Z).
$$
The chosen generator is therefore a local generator at every point and is a $\mathbb Z$-fundamental class.
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