 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: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 the classical and weak formulations are equivalent for continuously differentiable fields.