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Generalizations

  1. A functional can depend on more than one function. Then the functional derivative with respect to each argument is defined in the obvious way, each calculated while keeping the other function constant.
    Example.
    If

    equation560

    then

    eqnarray562

  2. The independent function can be a function of more than one variable.
    Example.
    Taking tex2html_wrap_inline900 to be a specific domain of integration in space, let

    equation570

    Then

    equation573

    just as in the one-dimensional case. F can also be written as tex2html_wrap_inline904 .

  3. A functional doesn't have to be a simple integral.
    Example.
    Take

    equation580

    Here, as always, the definition (5) gives a straightforward method for calculating the functional derivative tex2html_wrap_inline906 .