Laplace-Beltrami operator expression - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T09:16:20Z http://mathoverflow.net/feeds/question/105880 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/105880/laplace-beltrami-operator-expression Laplace-Beltrami operator expression Riadh 2012-08-29T20:25:59Z 2012-08-29T21:12:57Z <p>In the book Shape and Shape Theory of Kendall in p.147 I found the following expression of the Laplace-Beltrami operator: $\sum_i\left({v_i^2-\nabla_{v_i}v_i}\right)$ where $v_i$ are orthonormal tangent vectors. So please what does the exponent 2 stands for?</p> <p>Thank you</p> http://mathoverflow.net/questions/105880/laplace-beltrami-operator-expression/105882#105882 Answer by Benoît Kloeckner for Laplace-Beltrami operator expression Benoît Kloeckner 2012-08-29T21:12:57Z 2012-08-29T21:12:57Z <p>The $v_i$ are vector <em>fields</em>, and as such are derivations. The square usually means that you apply it twice (so, e.g. in the Euclidean space one can take $v_i=\frac{\partial}{\partial x_i}$ and its square is simply $\frac{\partial^2}{\partial x_i^2}$).</p>