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Functional derivative

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.


Functional chain rule

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 344 / 3 / g / Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 344 / 3 / b / Solution

Functional chain rule

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Functional chain rule

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