nextupprevious
Next:ReferencesUp:FunctionalsPrevious:Constraints

The delta function

Finally, let us return to the simplest functional of all,

 equation450

which just evaluates its argument y(x) at the specific point tex2html_wrap_inline1226 . What is its functional derivative tex2html_wrap_inline856 ? F only depends on the value of y at tex2html_wrap_inline1226 , so if we vary y at some other place, F will be unchanged. Thus

equation454

If we change y exactly at tex2html_wrap_inline1226 , however, F obviously changes.

An easy way to proceed is to rewrite (61) as an integral functional. Let's say there is some function tex2html_wrap_inline1246 that allows us to write

 equation459

What are the properties of tex2html_wrap_inline1248 ? Since F[y] has no dependence on y(x) for tex2html_wrap_inline1254 , we clearly have

    equation462       (64)


What is tex2html_wrap_inline1246 at tex2html_wrap_inline1258 ? If it were finite, the integral (63) would always be zero, because of the infinitesimal measure dx. Thus tex2html_wrap_inline1246 must be infinite at tex2html_wrap_inline1258 !

The actual value of tex2html_wrap_inline1266 will never really concern us. All we have to know is how to integrate with tex2html_wrap_inline1248 . In fact, we can take the following [which is just (61) and (63)] to be the definition of tex2html_wrap_inline1248 :

 equation470

In view of (63), we can state the following result, which must be used with care:

equation474

or, more compactly,

 equation478

Properly speaking, tex2html_wrap_inline1248 is not a function at all, since its infinite value takes us out of the usual domain of definition of functions. Mathematicians call it a distribution, a limit of a sequence of functions that really only has meaning in integral expressions such as (65). Let us evaluate (65) for the special case y(x)=1, choosing as well tex2html_wrap_inline1276 . We get

equation486

So the area under the tex2html_wrap_inline1278 function is 1 (even though its width is zero!). One possible realization of tex2html_wrap_inline1248 as a sequence of functions is the set of gaussians

equation488

Each tex2html_wrap_inline1282 has unit area, and tex2html_wrap_inline1282 becomes higher and narrower as tex2html_wrap_inline786 . Mathematicians will always be careful to insert tex2html_wrap_inline1288 into integrals like (65) and to evaluate the integral before taking the tex2html_wrap_inline786 limit.

Expressions such as (67) have meaning only when they are multiplied by some function of tex2html_wrap_inline1226 and integrated over tex2html_wrap_inline1226 ; then we are returned to the usual kind of functional derivative. At a physicist's level of rigor, however, (67) can be used to deal with chain rules and with higher functional derivatives, taking care to remember that the limit tex2html_wrap_inline786 may contain pitfalls.


nextupprevious
Next:ReferencesUp:FunctionalsPrevious:Constraints