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Applied Calculus of Variations for Engineers, 2nd Edition
book

Applied Calculus of Variations for Engineers, 2nd Edition

by Louis Komzsik
June 2014
Intermediate to advanced content levelIntermediate to advanced
233 pages
5h 42m
English
CRC Press
Content preview from Applied Calculus of Variations for Engineers, 2nd Edition
3
Multivariate functionals
3.1 Functionals with several functions
The variational problem of multiple dependent variables is posed as
I(y
1
,y
2
,...,y
n
)=
x
1
x
0
f(x, y
1
,y
2
,...,y
n
,y
1
,y
2
,...,y
n
)dx
with a pair of boundary conditions given for all functions:
y
i
(x
0
)=y
i,0
and
y
i
(x
1
)=y
i,1
for each i =1, 2,...,n. The alternative solutions are:
Y
i
(x)=y
i
(x)+
i
η
i
(x); i =1,...,n
with all the arbitrary auxiliary functions obeying the conditions:
η
i
(x
0
)=η
i
(x
1
)=0.
The variational problem becomes
I(
1
,...,
n
)=
x
1
x
0
f(x,...,y
i
+
i
η
i
,...,y
i
+
i
η
i
,...)dx,
whose derivative with respect to the auxiliary variables is
∂I
∂
i
=
x
1
x
0
∂f
∂
i
dx =0.
Applying the chain rule we get
∂f
∂
i
=
∂f
∂Y
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Publisher Resources

ISBN: 9781482253597