= Solution
If the fields are continuously differentiable and obey the pointwise <conservation law>, <integration by parts> against a compactly supported <test function> gives the weak identity, with no boundary terms. Conversely, integrate the weak identity by parts for a continuously differentiable field:
$$
\iint(\partial_t\mathbf P+\partial_x\mathbf Q)\cdot\boldsymbol\varphi\,dx\,dt=0
$$
for every <test function>. The <fundamental lemma of the calculus of variations>, applied to each component, makes the continuous integrand vanish at every point. Hence \b[the classical and weak formulations are equivalent for continuously differentiable fields].
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