Physics often deals with classical models where the dynamical variables are a collection of functions
{φ<sup>α</sup>}<sub>α</sub> over a d-dimensional space/spacetime manifold M where α is the "flavor" index. This involves functionals over the φs, functional derivatives, functional integrals, etc. From a functional point of view this is equivalent to working with an infinite-dimensional smooth manifold where its points are an assignment of a function for each α, and the procedure is in analogy with differential geometry where the coordinates for a point x of the manifold M are φ<sup>α</sup>(x).
In the DeWitt notation (named after theoretical physicist Bryce DeWitt), φ<sup>α</sup>(x) is written as φ<sup>i</sup> where i is now understood as an index covering both α and x.
So, given a smooth functional A, A<sub>,i</sub> stands for the functional derivative
:<math>A_{,i}[\varphi] \ \stackrel{\mathrm{def{=}\ \frac{\delta}{\delta \varphi^\alpha(x)}A[\varphi]</math>
as a functional of φ. In other words, a "1-form" field over the infinite dimensional "functional manifold".
In integrals, the Einstein summation convention is used. Alternatively,
:<math>A^i B_i \ \stackrel{\mathrm{def{=}\ \int_M \sum_\alpha A^\alpha(x) B_\alpha(x) d^dx</math>