Calculus of Variations and the Functional Derivative

[@friedrichwilhelmhufnagel3577]

Thank you for this crystal clear demonstration on the nature of functionals

[@KevinCassel]

Glad it was helpful!

[@RobertWF42]

Thanks for posting, Kevin! I'm coming to this topic from the field of statistics.

[@KevinCassel]

Glad it was helpful!

[@arkansh.h.1313]

Thank you for the great illustrations.
I hope if you can type the name of the book.

[@KevinCassel]

The book is in the description box below the video.

[@shreeniwaz]

Is epsilon equivalent to 'norm' of the function?

[@chadwinters4285]

I think the use of the term ϵ * η(x) is not so clear. What does du / dϵ look like geometrically?

[@KevinCassel]

Yes, this is confusing (which is why we prefer variational notation once we get used to it). Think of changes in epsilon as representing moving from one candidate function to another. Thus, du/deps represents the change in the function that we are seeking u(x) as we move from one function to another (via changes in eps). It is analogous to differential calculus, where du/dx represents the change in the function u(x) as we move from one point x to another.

[@keifonlee8342]

@@KevinCassel But in the presented notation, we got epsilon from no where, not like the du/dx where we know x is the independent variable. So the appearance of eps is not quite straight forward

[@CC-by3sn]

Does the functional derivative not depend on the x1 and x2 in the integral then?

[@KevinCassel]

The x1 and x2 define the domain for the differential equation given by the functional derivative, i.e. where the boundary conditions are defined.

[@exxzxxe]

Well done!

[@KevinCassel]

Thank you!

[@Ksnieup]

❤

[@KevinCassel]

Glad you like the videos.

[@martonzoldy1258]

Excuse me, I think it is the Dirac delta distribution instead of the Kronecker delta.