San José State University

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The Functional Derivative:
Its Natue and Use

The notion of the functional derivative arose out the Calculus of Variations. In the Calculus of Variations there is a functional such as

J[f] = ∫abL(x, f(x), f'(x))dx

which is to be maximized or minimized. It is found that this is achieved by the function f(x) which satisfies the Euler-Lagrange equation

(∂L/∂f) − d((∂L/∂f'))/dx
at all x from a to b

When

(∂L/∂f) − d((∂L/∂f'))/dx
is defined as the
functional derivative of J[f]
with respect to f
and denoted as
(δJ/δf)

then the condition for the optimal function f is expressed as

(δJ/δf) = 0

where 0 denotes the zero function from x=a to x=b.


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