basic_tools:variational_calculus:functional_derivative
| Next revision | Previous revision | ||
|
basic_tools:variational_calculus:functional_derivative [2018/03/10 17:30] iiqof created |
basic_tools:variational_calculus:functional_derivative [2018/03/15 14:56] (current) iiqof |
||
|---|---|---|---|
| Line 1: | Line 1: | ||
| + | |||
| + | |||
| ====== Functional Derivative ====== | ====== Functional Derivative ====== | ||
| Line 13: | Line 15: | ||
| where the last expression is equivalent, and will help us in the next definition, $\varepsilon$ represents the variation around the point $x$. | where the last expression is equivalent, and will help us in the next definition, $\varepsilon$ represents the variation around the point $x$. | ||
| - | For a functional, we will denote the derivative with $\delta$, as representation of variation. As we know how to derive functions of "numbers", we will transform our funtional of $q$ to a "normal" function for $\varepsilon$ with the map $F[q]\mapsto F[q+\varepsilon h]$ where $h$ is called a test function (analogous to a vector on multivariate calculus), then derive: | + | For a functional, we will denote the differential with $\delta$, as representation of variation. As we know how to derive functions of "numbers", we will transform our functional of $q$ to a "normal" function for $\varepsilon$ with the map $F[q]\mapsto F[q+\varepsilon \phi]$ where $\phi$ is called a test function (analogous to a vector on multivariate calculus), then derive: |
| $$ | $$ | ||
| Line 19: | Line 21: | ||
| $$ | $$ | ||
| + | The summand $\varepsilon \phi$ is called //variation// of $q$. | ||
| + | === Derivative of Integral Functionals === | ||
| + | For a [[basic_tools:variational_calculus:functional:integral_functional| integral functional ]], what interests us is the derivative with respect a function. This is INCOMPLETE; WILL DO LATER | ||
| - | __** Formal Definitions **__ | + | |
| + | ==== Formal Definitions ==== | ||
| Frechet derivative on Banach spaces, and more generally Gatheaux derivative on locally convex spaces ... | Frechet derivative on Banach spaces, and more generally Gatheaux derivative on locally convex spaces ... | ||
| Line 28: | Line 34: | ||
| - | __** References **__ | + | <tabbox Exercises> |
| + | |||
| + | 1. | ||
| + | </tabbox> | ||
basic_tools/variational_calculus/functional_derivative.1520699441.txt.gz · Last modified: 2018/03/10 16:30 (external edit)
Page Tools
Except where otherwise noted, content on this wiki is licensed under the following license: CC Attribution-Share Alike 4.0 International
