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## Integrals

a) Let $A=(a_{ij})_{i,j=1}^n$ be a real matrix. Show that

$\displaystyle \int_{|x|<1} (x,Ax) dx = \frac{\omega_n}{n(n+2)} tr(A),$ where $(\cdot ,\cdot)$ is the usual dot product and $\omega_n$ is the area  of the unit sphere.

b) Show that for all $u \in C_0^2 (\Bbb{R}^n)$ we have

$\displaystyle \int_{\Bbb{R}^n} (\Delta u)^2 dx =\sum_{i,j=1}^n \int_{\Bbb{R}^n} |D_{ij}u|^2 dx.$

PHD Iowa (6101)

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