The functional derivative defined using the delta function

Start with the functional derivative expression:

$$ \frac{dF[f + \epsilon \eta]}{d\epsilon} \bigg|_{\epsilon = 0} $$

Replace the function \( \eta(x) \) with a shifted delta function \( \delta(x - x_0)\) and also express \( \epsilon \eta(x)\) as \( \epsilon \delta(x - x_0) \):

$$ \frac{dF[f + \epsilon \delta(x - x_0)]}{d\epsilon} \bigg|_{\epsilon = 0} $$

Now, the next step is to change the derivative with respect to \( \epsilon \) into a limit:

$$ \lim_{\epsilon \rightarrow 0} \frac{1}{\epsilon} \left[F[f + \epsilon \delta(x - x_0)] - F[f]\right] $$

Finally, we can rewrite the limit using integrals:

$$ \lim_{\epsilon \rightarrow 0} \frac{1}{\epsilon} \left[\int_{-1}^1[f(x) + \epsilon \delta(x - x_0)]dx - \int_{-1}^1 f(x)dx\right] $$

Now, let's understand the role of the delta function in this expression:

The delta function, often denoted as \( \delta(x - x_0) \), is a distribution that behaves like a generalized function and is defined such that:

$$ \int_{-\infty}^{\infty} f(x) \delta(x - x_0) dx = f(x_0) $$

where f(x) is a test function (a well-behaved smooth function) and \( x_0 \) is a point in the domain.

In the context of the functional derivative, the delta function \( \delta(x - x_0) \) allows us to isolate the effect of a small localized perturbation at the point \( x_0 \) in the function f(x). When we add the term \( \epsilon \delta(x - x_0) \) to the function f(x) and take the functional derivative with respect to \( \epsilon \), we are effectively probing how the functional F[f] responds to a localized change (perturbation) in the function f(x) at the point \( x_0 \).

By taking the limit as \( \epsilon \) approaches 0, we are considering an infinitesimally small perturbation around \( x_0 \). This allows us to examine the "instantaneous" response of F[f] to a localized change at \( x_0 \), which is encoded in the functional derivative. The expression involving the limit of the integral is an equivalent way of expressing this behavior.

The limit defined using the delta function is not considered to be mathematically rigorous, but is sufficiently acceptable as a mathematical aid for calculating the functional derivative in many cases.