Ngô Quốc Anh

August 9, 2011

Stereographic projection, 5

Filed under: PDEs, Riemannian geometry — Tags: — Ngô Quốc Anh @ 14:39

I put here several common formulas that are needed when we work on stereographic projection. First we try to calculate

\displaystyle\Delta \left( {{{\left( {\frac{2}{{1 + |x{|^2}}}} \right)}^{n - 2}}} \right).

We do step by step.

Step 1. A direct computation leads us to

\displaystyle\begin{gathered} {\partial _i}\left( {{{\left( {\frac{2}{{1 + |x{|^2}}}} \right)}^{n - 2}}} \right) = (n - 2){\left( {\frac{2}{{1 + |x{|^2}}}} \right)^{n - 3}}{\partial _i}\left( {\frac{2}{{1 + |x{|^2}}}} \right) \hfill \\ \qquad= (n - 2){\left( {\frac{2}{{1 + |x{|^2}}}} \right)^{n - 3}}\frac{{ - 4{x_i}}}{{{{(1 + |x{|^2})}^2}}}. \hfill \\ \end{gathered}

Therefore,

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