# Introduction to Mathematical Physics/Energy in continuous media/Exercises

Exercice: Find the equation evolution for a rope clamped between two walls.

Exercice: {{{1}}}

Exercice: Give the expression of the deformation energy of a smectic (see section secristliquides for the description of smectic) whose i${\displaystyle ^{t}h}$ layer's state is described by surface ${\displaystyle u_{i}(x,y)}$,

Exercice: Consider a linear, homogeneous, isotrope material. Electric susceptibility ${\displaystyle \epsilon }$ introduced at section secchampdslamat allows for such materials to provide ${\displaystyle D}$ from ${\displaystyle E}$ by simple convolution:

eqexsusc

${\displaystyle D=\epsilon *E.}$

where ${\displaystyle *}$ represents a temporal convolution. To obey to the causality principle distribution ${\displaystyle \epsilon }$ has to have a positive support. Indeed, ${\displaystyle D}$ can not depend on the future values of ${\displaystyle E}$. Knowing that the Fourier transform of function "sign of ${\displaystyle t}$" is ${\displaystyle C.V_{p}(1/x)}$ where ${\displaystyle C}$ is a normalization constant and ${\displaystyle V_{p}(1/X)}$ is the principal value of ${\displaystyle 1/x}$ distribution, give the relations between the real part and imaginary part of the Fourier transform of ${\displaystyle \epsilon }$. These relations are known in optics as Krammers--Kr\"onig relations\index{Krammers--Kr\"onig relations}.