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Functional Derivative on Vector Field (Definition)

0

$begingroup$

Given a set $mathcalM$ of functions $mathcalX to mathbbR$, a functional $J: mathcalM to mathbbR$ and $f, phi in mathcalM$, the functional derivative of $J$ at $f$, $D_f(J)$ is the (unique) element of $M$ such that
$$
int_mathcalX D_f(J)(x)phi(x)dx = fracddepsilon J[g + epsilon phi] vert_epsilon = 0.
$$

I am interested in a case in which the functions are $mathcalX to mathbbR^N$. Do we simply exchange the above for
$$
int_mathcalX D_f(J)(x)^Tphi(x)dx = fracddepsilon J[g + epsilon phi] vert_epsilon = 0?
$$

calculus functional-analysis derivatives definition calculus-of-variations

share|cite|improve this question

edited Apr 2 at 21:03

user27182

asked Apr 2 at 14:48

user27182user27182

1,042727

$endgroup$

add a comment | 

0

$begingroup$

Given a set $mathcalM$ of functions $mathcalX to mathbbR$, a functional $J: mathcalM to mathbbR$ and $f, phi in mathcalM$, the functional derivative of $J$ at $f$, $D_f(J)$ is the (unique) element of $M$ such that
$$
int_mathcalX D_f(J)(x)phi(x)dx = fracddepsilon J[g + epsilon phi] vert_epsilon = 0.
$$

I am interested in a case in which the functions are $mathcalX to mathbbR^N$. Do we simply exchange the above for
$$
int_mathcalX D_f(J)(x)^Tphi(x)dx = fracddepsilon J[g + epsilon phi] vert_epsilon = 0?
$$

calculus functional-analysis derivatives definition calculus-of-variations

share|cite|improve this question

edited Apr 2 at 21:03

user27182

asked Apr 2 at 14:48

user27182user27182

1,042727

$endgroup$

add a comment | 

$begingroup$

Given a set $mathcalM$ of functions $mathcalX to mathbbR$, a functional $J: mathcalM to mathbbR$ and $f, phi in mathcalM$, the functional derivative of $J$ at $f$, $D_f(J)$ is the (unique) element of $M$ such that
$$
int_mathcalX D_f(J)(x)phi(x)dx = fracddepsilon J[g + epsilon phi] vert_epsilon = 0.
$$

I am interested in a case in which the functions are $mathcalX to mathbbR^N$. Do we simply exchange the above for
$$
int_mathcalX D_f(J)(x)^Tphi(x)dx = fracddepsilon J[g + epsilon phi] vert_epsilon = 0?
$$

calculus functional-analysis derivatives definition calculus-of-variations

share|cite|improve this question

edited Apr 2 at 21:03

user27182

asked Apr 2 at 14:48

user27182user27182

1,042727

$endgroup$

share|cite|improve this question

edited Apr 2 at 21:03

user27182

asked Apr 2 at 14:48

user27182user27182

1,042727

share|cite|improve this question

edited Apr 2 at 21:03

user27182

asked Apr 2 at 14:48

user27182user27182

1,042727

asked Apr 2 at 14:48

user27182user27182

1,042727

asked Apr 2 at 14:48

user27182user27182

1,042727