Step 1: What is a Functional?

A functional is a mapping from functions to numbers. While a regular function takes numbers as input and outputs numbers, a functional takes an entire function (or functions) as input and outputs a single number.

Examples:

• The arc length of a curve \(y(x)\): \[L[y] = \int \sqrt{1 + \left(\frac{dy}{dx}\right)^2} dx\]
• The action in mechanics: \[S[q(t)] = \int L(q, \dot{q}, t) dt\]
• The energy of a field configuration: \[E[\varphi] = \int \frac{1}{2}(\nabla\varphi)^2 d^3x\]

Notation:

We write \(R[\rho, \Phi]\) with square brackets to indicate \(R\) is a functional that takes the entire functions \(\rho(x)\) and \(\Phi(x)\) as inputs and returns a single number.

Key property:

Small changes in the input functions lead to variations:

\[\delta R = \int \frac{\delta R}{\delta \rho} \delta\rho \, d^3x + \int \frac{\delta R}{\delta \Phi} \delta\Phi \, d^3x\]

where \(\frac{\delta R}{\delta \rho}\) and \(\frac{\delta R}{\delta \Phi}\) are the functional derivatives - they tell us how the output changes when we slightly modify the input functions at each point.

Mini-Glossary

Symbol	Name	Meaning	Value / Units	Metaphor
\(R[\rho, \Phi]\)	redundancy functional	mapping from functions to a number	dimensionless	"Total information capacity score"
\(\rho(x)\)	matter density	energy density at position x	energy/volume	"How much stuff is here"
\(\Phi(x)\)	time potential	logarithm of lapse function	dimensionless	"Altitude of time"
\(\frac{\delta R}{\delta \rho}\)	functional derivative	how R changes with local density	volume/energy	"Sensitivity to adding matter"
\(L[y]\), \(S[q]\), \(E[\varphi]\)	example functionals	arc length, action, energy	various	"Different ways to score functions"