Physics often deals with classical models where the dynamical variables are a collection of functions {φα}α over a d-dimensional space/spacetime manifold M where α is the "flavor" index. This involves functionals over the φ's, 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 φα(x). In the DeWitt notation (named after theoretical physicist Bryce DeWitt), φα(x) is written as φi where i is now understood as an index covering both α and x. So, given a smooth functional A, A,i stands for the functional derivative
A , i [ φ ] = d e f δ δ φ α ( x ) A [ φ ] {\displaystyle A_{,i}[\varphi ]\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\delta }{\delta \varphi ^{\alpha }(x)}}A[\varphi ]}
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,
A i B i = d e f ∫ M ∑ α A α ( x ) B α ( x ) d d x {\displaystyle A^{i}B_{i}\ {\stackrel {\mathrm {def} }{=}}\ \int _{M}\sum _{\alpha }A^{\alpha }(x)B_{\alpha }(x)d^{d}x}
References Kiefer, Claus (April 2007). Quantum gravity (hardcover) (2nd ed.). Oxford University Press. p. 361. ISBN 978-0-19-921252-1.