basic_tools:variational_calculus:functional_derivative
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basic_tools:variational_calculus:functional_derivative [2018/03/10 17:32] iiqof Added Exercicises section |
basic_tools:variational_calculus:functional_derivative [2018/03/15 14:56] (current) iiqof |
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| ====== Functional Derivative ====== | ====== Functional Derivative ====== | ||
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| 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: |
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| + | The summand $\varepsilon \phi$ is called //variation// of $q$. | ||
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| + | === 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 ... | ||
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