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Center of mass of a surface

Consider a two dimensional surface $ \color{white} \mathfrak{S}$, choose a system of coordinates and define a position vector $ \color{white} \vec{r}_{\mathfrak{S}}$ of its center of mass as following
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where $ \color{white} \vec{r}$ is a position vector of an element $ \color{white} \mathrm{d} S$.