Functional Derivative
Overview
Definition
{% \frac{\delta F}{ \delta f} = lim_{\epsilon \rightarrow 0} \frac{F[f(x') + \epsilon \delta(x -x')] - F[f(x')]}{\epsilon} %}
where {% \delta(x) %} is the
Dirac Delta Function
Example 1
{% \frac{\delta I[f]}{\delta f(x_0)} = lim_{\epsilon \rightarrow 0} \frac{1}{\epsilon} \int_{-1}^1 [f(x) + \epsilon \delta(x-x_0)]dx - \int_{-1}^1 f(x)dx %}
{% = \int_{-1}^1 \delta(x-x_0)dx %}
which is equal to 1 for {% -1 \leq x_0 \leq 1 %} and 0 otherwise.
Example 2
{% H[f] = \int_a^b g[f(x)] dx %}
{% \frac{\delta H[f]}{\delta f(x_0)} = lim_{\epsilon \rightarrow 0} [\int g[f(x) + \epsilon \delta(x-x_0)]dx - \int g[f(x)]dx] %}
{% = lim_{\epsilon \rightarrow 0} [\int g[f(x)] + \epsilon \delta(x-x_0) g'[f(x)] dx - \int g[f(x)]dx] %}
{% = \int \delta(x-x_0) g'[f(x)] dx %}
{% = g'[f(x_0)] %}
