and theoretical physics
, the functional derivative
is a generalization of the directional derivative
. The difference is that the latter differentiates in the direction of a vector, while the former differentiates in the direction of a function. Both of these can be viewed as extensions of the usual calculus derivative
Two possible, restricted definitions suitable for certain computations are given here. There are more general definitions of functional derivatives relying on ideas from functional analysis, such as the Gâteaux derivative.
Given a manifold M representing (continuous/smooth/with certain boundary conditions/etc.) functions φ and a functional F defined as
the functional derivative of F, denoted , is a distribution such that for all test functions f,
Sometimes physicists write the definition in terms of a limit and the Dirac delta function, δ:
The definition of a functional derivative may be made much more mathematically precise and formal by defining the space of functions
more carefully. For example, when the space of functions is a Banach space
, the functional derivative becomes known as the Fréchet derivative
, while one uses the Gâteaux derivative
on more general locally convex spaces
. Note that the well-known Hilbert spaces
are special cases of Banach spaces
. The more formal treatment allows many theorems from ordinary calculus
to be generalized to corresponding theorems in functional analysis
, as well as numerous new theorems to be stated.
Relationship between the mathematical and physical definitions
The mathematicians' definition and the physicists' definition of the functional derivative differ only in the physical interpretation. Since the mathematical definition is based on a relationship that holds for all test functions f
, it should also hold when f
is chosen to be a specific function. The only handwaving difficulty is that specific function was chosen to be a delta function
---which is not a valid test function.
In the mathematical definition, the functional derivative describes how the entire functional, , changes as a result of a small change in the function . Observe that the particular form of the change in is not specified. The physics definition, by contrast, employs a particular form of the perturbation --- namely, the delta function --- and the 'meaning' is that we are varying only about some neighborhood of . Outside of this neighborhood, there is no variation in .
Often, a physicist wants to know how one quantity, say the electric potential at position , is affected by changing another quantity, say the density of electric charge at position . The potential at a given position, is a functional of the density. That is, given a particular density function and a point in space, one can compute a number which represents
the potential of that point in space due to the specified density function. Since we are interested in how this number varies across all points in space, we treat the potential as a function of . To wit,