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Mass conservation in a deformed membrane in cylindrical coordinates

At time $t$ after impact, we let $u$ and $v$ be the in-plane and normal, out-of-plane components, respectively, of the displacement of this point, but its cylindrical angle $\phi$ remains the same. … However, I am trying to demonstrate it in a more formal way and to do that I wrote the displacement vector $\vec{u} = (u,0,v)$ in the cylindrical space and that's in order to obtain the expression of the …
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