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MultiFunction Variation
Suppose that we wish to maximize or minimize the functional

(707) 
Here, the integrand is now a functional of the independent
functions , for . A fairly straightforward extension of the
analysis in Section 10.2 yields separate EulerLagrange equations,

(708) 
for , which determine the functions . If does not
explicitly depend on the function then the th EulerLagrange
equation simplifies to

(709) 
Likewise, if does not explicitly depend on then all EulerLagrange equations simplify to

(710) 
for .
Richard Fitzpatrick
20110331