Physics often deals 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. 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 *φ*^{α}(*x*).

In the **DeWitt notation** (named after theoretical physicist Bryce DeWitt), φ^{α}(*x*) is written as φ^{i} 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

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,

## References

- Kiefer, Claus (April 2007).
*Quantum gravity*(hardcover) (2nd ed.). Oxford University Press. p. 361. ISBN 978-0-19-921252-1.