Derivative of a functional with respect to another functional












0














I am trying to make sense of functional derivatives and have a couple of questions bothering me:





  1. Let $F[X]$ be a functional of $X(t)$ and $G[X]$ another functional of $X(t)$.



    By chain rule in the continum I intuitively guess that the derivative of the functional F[X] with respect to the functional G[X] is:$$frac{delta F}{delta G} = int dt frac{delta F[X]}{delta X(t)} frac{delta X(t)}{delta G[X]} = int dt frac{delta F[X]}{delta X(t)} left(frac{delta G[X]}{delta X(t)} right)^{-1} $$
    Does this make any sense at all? Is there such thing as a derivative of a functional with respect to another functional given they are both dependent on the same underlying function X(t)?



  2. How would one calculate the functional derivatives $frac{delta F[X]}{delta X(t)}$ and $frac{delta G[X]}{delta X(t)}$? I came across Gateaux derivative and It seems to be similar to a directional derivative. Does it matter what direction is being chosen to evaluate the final functional derivative?











share|cite|improve this question




















  • 1




    I am not sure what you are asking. Please rewrite.
    – Will M.
    Dec 11 '18 at 5:14










  • Thanks Will, just edited please let me know if it's still not clear.
    – ZeroCool
    Dec 11 '18 at 15:44










  • OK. It still makes no sense. However, two points to bring. (1) Frechet derivative is simply the usual derivative on learns from calculus 2 (at least in places like the one I studied where math rigour is imposed over being able to perform calculations) and (2) Gateaux derivatives are directional derivatives and, as always, they depend on the chosen direction. If you want to know the derivatives of a composition $g circ f,$ then simply apply the chain rule $g'(f(a)) circ f'(a).$ If you want to know how to calculate $f'$ or $g',$ you need to know the exact form first an apply rules then.
    – Will M.
    Dec 11 '18 at 18:05
















0














I am trying to make sense of functional derivatives and have a couple of questions bothering me:





  1. Let $F[X]$ be a functional of $X(t)$ and $G[X]$ another functional of $X(t)$.



    By chain rule in the continum I intuitively guess that the derivative of the functional F[X] with respect to the functional G[X] is:$$frac{delta F}{delta G} = int dt frac{delta F[X]}{delta X(t)} frac{delta X(t)}{delta G[X]} = int dt frac{delta F[X]}{delta X(t)} left(frac{delta G[X]}{delta X(t)} right)^{-1} $$
    Does this make any sense at all? Is there such thing as a derivative of a functional with respect to another functional given they are both dependent on the same underlying function X(t)?



  2. How would one calculate the functional derivatives $frac{delta F[X]}{delta X(t)}$ and $frac{delta G[X]}{delta X(t)}$? I came across Gateaux derivative and It seems to be similar to a directional derivative. Does it matter what direction is being chosen to evaluate the final functional derivative?











share|cite|improve this question




















  • 1




    I am not sure what you are asking. Please rewrite.
    – Will M.
    Dec 11 '18 at 5:14










  • Thanks Will, just edited please let me know if it's still not clear.
    – ZeroCool
    Dec 11 '18 at 15:44










  • OK. It still makes no sense. However, two points to bring. (1) Frechet derivative is simply the usual derivative on learns from calculus 2 (at least in places like the one I studied where math rigour is imposed over being able to perform calculations) and (2) Gateaux derivatives are directional derivatives and, as always, they depend on the chosen direction. If you want to know the derivatives of a composition $g circ f,$ then simply apply the chain rule $g'(f(a)) circ f'(a).$ If you want to know how to calculate $f'$ or $g',$ you need to know the exact form first an apply rules then.
    – Will M.
    Dec 11 '18 at 18:05














0












0








0







I am trying to make sense of functional derivatives and have a couple of questions bothering me:





  1. Let $F[X]$ be a functional of $X(t)$ and $G[X]$ another functional of $X(t)$.



    By chain rule in the continum I intuitively guess that the derivative of the functional F[X] with respect to the functional G[X] is:$$frac{delta F}{delta G} = int dt frac{delta F[X]}{delta X(t)} frac{delta X(t)}{delta G[X]} = int dt frac{delta F[X]}{delta X(t)} left(frac{delta G[X]}{delta X(t)} right)^{-1} $$
    Does this make any sense at all? Is there such thing as a derivative of a functional with respect to another functional given they are both dependent on the same underlying function X(t)?



  2. How would one calculate the functional derivatives $frac{delta F[X]}{delta X(t)}$ and $frac{delta G[X]}{delta X(t)}$? I came across Gateaux derivative and It seems to be similar to a directional derivative. Does it matter what direction is being chosen to evaluate the final functional derivative?











share|cite|improve this question















I am trying to make sense of functional derivatives and have a couple of questions bothering me:





  1. Let $F[X]$ be a functional of $X(t)$ and $G[X]$ another functional of $X(t)$.



    By chain rule in the continum I intuitively guess that the derivative of the functional F[X] with respect to the functional G[X] is:$$frac{delta F}{delta G} = int dt frac{delta F[X]}{delta X(t)} frac{delta X(t)}{delta G[X]} = int dt frac{delta F[X]}{delta X(t)} left(frac{delta G[X]}{delta X(t)} right)^{-1} $$
    Does this make any sense at all? Is there such thing as a derivative of a functional with respect to another functional given they are both dependent on the same underlying function X(t)?



  2. How would one calculate the functional derivatives $frac{delta F[X]}{delta X(t)}$ and $frac{delta G[X]}{delta X(t)}$? I came across Gateaux derivative and It seems to be similar to a directional derivative. Does it matter what direction is being chosen to evaluate the final functional derivative?








functional-analysis frechet-derivative gateaux-derivative






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 11 '18 at 15:43

























asked Nov 28 '18 at 0:01









ZeroCool

28619




28619








  • 1




    I am not sure what you are asking. Please rewrite.
    – Will M.
    Dec 11 '18 at 5:14










  • Thanks Will, just edited please let me know if it's still not clear.
    – ZeroCool
    Dec 11 '18 at 15:44










  • OK. It still makes no sense. However, two points to bring. (1) Frechet derivative is simply the usual derivative on learns from calculus 2 (at least in places like the one I studied where math rigour is imposed over being able to perform calculations) and (2) Gateaux derivatives are directional derivatives and, as always, they depend on the chosen direction. If you want to know the derivatives of a composition $g circ f,$ then simply apply the chain rule $g'(f(a)) circ f'(a).$ If you want to know how to calculate $f'$ or $g',$ you need to know the exact form first an apply rules then.
    – Will M.
    Dec 11 '18 at 18:05














  • 1




    I am not sure what you are asking. Please rewrite.
    – Will M.
    Dec 11 '18 at 5:14










  • Thanks Will, just edited please let me know if it's still not clear.
    – ZeroCool
    Dec 11 '18 at 15:44










  • OK. It still makes no sense. However, two points to bring. (1) Frechet derivative is simply the usual derivative on learns from calculus 2 (at least in places like the one I studied where math rigour is imposed over being able to perform calculations) and (2) Gateaux derivatives are directional derivatives and, as always, they depend on the chosen direction. If you want to know the derivatives of a composition $g circ f,$ then simply apply the chain rule $g'(f(a)) circ f'(a).$ If you want to know how to calculate $f'$ or $g',$ you need to know the exact form first an apply rules then.
    – Will M.
    Dec 11 '18 at 18:05








1




1




I am not sure what you are asking. Please rewrite.
– Will M.
Dec 11 '18 at 5:14




I am not sure what you are asking. Please rewrite.
– Will M.
Dec 11 '18 at 5:14












Thanks Will, just edited please let me know if it's still not clear.
– ZeroCool
Dec 11 '18 at 15:44




Thanks Will, just edited please let me know if it's still not clear.
– ZeroCool
Dec 11 '18 at 15:44












OK. It still makes no sense. However, two points to bring. (1) Frechet derivative is simply the usual derivative on learns from calculus 2 (at least in places like the one I studied where math rigour is imposed over being able to perform calculations) and (2) Gateaux derivatives are directional derivatives and, as always, they depend on the chosen direction. If you want to know the derivatives of a composition $g circ f,$ then simply apply the chain rule $g'(f(a)) circ f'(a).$ If you want to know how to calculate $f'$ or $g',$ you need to know the exact form first an apply rules then.
– Will M.
Dec 11 '18 at 18:05




OK. It still makes no sense. However, two points to bring. (1) Frechet derivative is simply the usual derivative on learns from calculus 2 (at least in places like the one I studied where math rigour is imposed over being able to perform calculations) and (2) Gateaux derivatives are directional derivatives and, as always, they depend on the chosen direction. If you want to know the derivatives of a composition $g circ f,$ then simply apply the chain rule $g'(f(a)) circ f'(a).$ If you want to know how to calculate $f'$ or $g',$ you need to know the exact form first an apply rules then.
– Will M.
Dec 11 '18 at 18:05















active

oldest

votes











Your Answer





StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");

StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});

function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});


}
});














draft saved

draft discarded


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3016507%2fderivative-of-a-functional-with-respect-to-another-functional%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown






























active

oldest

votes













active

oldest

votes









active

oldest

votes






active

oldest

votes
















draft saved

draft discarded




















































Thanks for contributing an answer to Mathematics Stack Exchange!


  • Please be sure to answer the question. Provide details and share your research!

But avoid



  • Asking for help, clarification, or responding to other answers.

  • Making statements based on opinion; back them up with references or personal experience.


Use MathJax to format equations. MathJax reference.


To learn more, see our tips on writing great answers.





Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


Please pay close attention to the following guidance:


  • Please be sure to answer the question. Provide details and share your research!

But avoid



  • Asking for help, clarification, or responding to other answers.

  • Making statements based on opinion; back them up with references or personal experience.


To learn more, see our tips on writing great answers.




draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3016507%2fderivative-of-a-functional-with-respect-to-another-functional%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown





















































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown

































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown







Popular posts from this blog

Probability when a professor distributes a quiz and homework assignment to a class of n students.

Aardman Animations

Are they similar matrix