Relevant to the dynamics of a beam, consider the fourth order linear differential operator.

$\displaystyle L=\frac{d^4}{dx^4}$

a. Show that $\displaystyle uLv - vLu$ is an exact differential. It is a differential of an expression with four terms, each term a product of $\displaystyle u$ and $\displaystyle v$ or their derivatives.

b. Evaluate $\displaystyle \int \begin{array}{cc}1\\0\end{array} (uLv - vLu) dx$ in terms of boundary data for any functions $\displaystyle u$ and $\displaystyle v$.

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