Functional chain rule
= Functional chain rule
Along a differentiable path of fields $\phi(x,t)$, a differentiable functional obeys
$$
\frac{d}{dt}F[\phi(t)]
=\int dx\,
\frac{\delta F}{\delta\phi(x)}\,
\partial_t\phi(x,t).
$$
Integrating in time expresses the functional change as the line integral of its functional derivative along the path.