DeWitt notation

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In physics, we often deal with classical models where the dynamical variables are a collection of functions {φ^α}_α over a d-dimensional space/spacetime manifold M where α is the "flavor" index. We then deal with functionals over the φ's, functional derivatives, functional integrals, etc. If we choose to take a functional point of view, it's as if we are working with an infinite-dimensional smooth manifold where its points are an assignment of a function for each α and we can proceed in analogy with differential geometry where the coordinates are φ^α(x) where x is a point of M.

In the deWitt notation we write φ^α(x) as φⁱ where i is now understood as an index covering both α and x.

So, if we have a smooth functional A, A_,i stands for the functional derivative

$\frac{\delta}{\delta \phi^\alpha(x)}A[\phi]$