Looking for a reference for the laplacian operator - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T04:47:32Z http://mathoverflow.net/feeds/question/10272 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/10272/looking-for-a-reference-for-the-laplacian-operator Looking for a reference for the laplacian operator Anirbit 2009-12-31T14:15:31Z 2010-03-01T08:03:40Z <p>Can anyone give me a reference which explain the derivation of the partial differential operator expression for the laplacian on the euclidean n-dimensional space and on $S^n$ ?</p> <p>One generally writes the laplacian on the n-dim euclidean space as a sum of a operator on the radial coordinate and $\frac{1}{r^2}$ times the laplacian on $S^n$. </p> <p>And very often the laplacian on $S^n$ is written through a recursion relation. </p> <p>I am looking for a reference which shows me the derivations of these. </p> http://mathoverflow.net/questions/10272/looking-for-a-reference-for-the-laplacian-operator/10274#10274 Answer by Akhil Mathew for Looking for a reference for the laplacian operator Akhil Mathew 2009-12-31T14:29:33Z 2009-12-31T14:45:00Z <p>The Laplacian can be defined on any Riemannian manifold as div grad. Here grad f for f a smooth function is the vector field dual to the 1-form df via the bilinear form of the metric. Div of a vector field X corresponds to taking the covariant derivative $\nabla X$, which is a (1,1) tensor, and taking the trace of that. In local coordinates one can give a formula using the symbols for the metric, which should yield what you are looking for. </p> <p>Another way to define div is to take the Lie derivative of the volume form: that is, $L_X V = (div X) V$. The volume form depends on an orientation, which can be locally chosen. This way is actually probably easier for computing in local coordinates since you don't need to worry about a covariant derivative or Christoffel symbols.</p> <p>For a reference, see e.g. Taylor's <em>Partial Differential Equations,</em> Vol. 1. In Folland's <em>Introduction to Partial Differential Equations,</em> there isn't much about Riemannian manifolds, but Folland does talk about how the Laplacian changes with respect to new coordinates.</p> http://mathoverflow.net/questions/10272/looking-for-a-reference-for-the-laplacian-operator/10287#10287 Answer by Steve Huntsman for Looking for a reference for the laplacian operator Steve Huntsman 2009-12-31T16:39:05Z 2009-12-31T17:12:08Z <p>In $\mathbb{R}^n$ consider $f = f(r)$. Writing $\partial_j \equiv \partial/\partial x_j$ and $\partial_r \equiv \partial/\partial_r$, etc., we have that $\partial_j r = x_j/r$, so</p> <p>$\partial_j f = (\partial_r f)(x_j r^{-1})$</p> <p>and</p> <p>$\partial_{jj} f = (\partial_{rr} f)(x_j^2 r^{-2}) + (\partial_r f)(r^{-1} - x_j^2 r^{-3})$.</p> <p>Summing over $j$ and comparing with the Cartesian expression for $\Delta$ gives the decomposition into radial and spherical operators. To be explicit you should consider $f = f(r, \omega)$, where $\omega \in S^{n-1}$.</p> <p>For a more general case, see the end of Chapter 2 of <em>The Laplacian on a Riemannian manifold: an introduction to analysis on manifolds</em> by Rosenberg. Unfortunately some of the relevant section (the Laplacian in exponential coordinates) is blocked both from Amazon and Google:</p> <p><a href="http://books.google.com/books?id=gzJ6Vn0y7XQC&amp;dq=laplacian+on+a+riemannian+manifold&amp;source=gbs_navlinks_s" rel="nofollow">http://books.google.com/books?id=gzJ6Vn0y7XQC&amp;dq=laplacian+on+a+riemannian+manifold&amp;source=gbs_navlinks_s</a></p> http://mathoverflow.net/questions/10272/looking-for-a-reference-for-the-laplacian-operator/10289#10289 Answer by Deane Yang for Looking for a reference for the laplacian operator Deane Yang 2009-12-31T16:55:45Z 2009-12-31T17:16:47Z <p>The Laplacian originates from physics. In particular, it arises as the linear differential operator in the Euler-Lagrange equation for the functional $f \mapsto E[f] = \int |\nabla f|^2$. You can derive formulas for the Laplacian on either Euclidean space or the unit sphere by differentiating this functional with respect to $f$ and determining the condition for a critical point.</p> <p>You can figure out the relationship between the Euclidean and spherical Laplacians by observing that in polar co-ordinates, $|\nabla f|^2 = |\partial_rf|^2 + r^2|\partial_\theta f|^2$, where $\nabla$ is the Euclidean gradient and $\partial_\theta$ is the spherical gradient.</p> <p>The recurrence relation for the spherical Laplacian arises from the observation in polar co-ordinates the $(n-1)$-dimensional spherical gradient can be written as $|\partial_\theta f|^2 = |\partial_\phi f|^2 + (\sin\phi)^2|\partial'_\theta f|^2$, where $\phi \in [0,\pi)$ is the co-ordinate giving the angle between a point and $e_n$, $\theta \in S^{n-2}$, and $\partial_\theta f$ is the $(n-2)$-dimensional spherical gradient.</p> <p>These formulas, at least for dimensions 2 and 3, can be found in most textbooks on electromagnetic theory or mathematical physics. The trick is to use the same derivation given in these books but recast them in a more abstract arbitrary dimension form.</p> <p>If you want to work things out using Riemannian geometry, I recommend using stereographic co-ordinates on the unit sphere.</p> http://mathoverflow.net/questions/10272/looking-for-a-reference-for-the-laplacian-operator/10571#10571 Answer by Anirbit for Looking for a reference for the laplacian operator Anirbit 2010-01-03T09:29:24Z 2010-01-03T09:42:13Z <p>So I wrote up this small derivation drawing insights from the answer by Deane Yang and Steve Huntsman, </p> <p>With respect to the Riemann-Christoffel connection on a riemannian manifold the laplacian on that manifold will have the form $$\nabla ^2 \phi = \frac{1}{\sqrt{g}} \partial _{\mu} \left [ \sqrt{g} g^{\mu \nu} \partial _{\nu} \phi \right ]$$</p> <p>where $g$ is the the determinant of the metric on the manifold and $\phi$ is some smooth scalar function on the manifold. </p> <p>On can write the line element on $S^n \subset \mathbb{R}^{n+1}$ as, </p> <p><code>$d\Omega _n ^2 = d\theta _1 ^2 + sin^2 \theta_1 d\theta _2 ^2 + sin^2 \theta _1 sin^2 \theta_2 d\theta _3 ^3 +...+sin^2\theta _1 sin^2 \theta_2...sin^2 \theta_{n-2} sin^2 \theta_{n-1} d\theta _n ^2$</code> </p> <p>Then the line element on $\mathbb{R}^{n+1}$ in polar coordinates can be written as,</p> <p>$$ds^2 = dr^2 + r^2 d\Omega _n ^2$$ </p> <p>and <code>$g_{\mathbb{R}^{n+1}} = r^{2n}g_{_{S^n}}$</code> where <code>$g_{_{S^n}} = (sin^2 \theta _1)^{n-1}(sin^2 \theta_2)^{n-2}...(sin^2 \theta _{n-2})^2(sin^2 \theta_{n-1})^1$</code></p> <p>Therefore since the metric is diagonal <code>$\nabla _{\mathbb{R}^{n+1}} ^2 \phi = \frac{1}{r^n \sqrt{g_{_{S^n}} }} \partial _{\mu} \left [r^n \sqrt{g_{_{S^n}} } g^{\mu \mu}_{\mathbb{R}^{n+1}} \partial _{\mu} \phi \right ]$</code></p> <p><code>$=\frac{1}{r^n \sqrt{g_{_{S^n}} }} \partial _{r} \left [r^n \sqrt{g_{_{S^n}} } \partial _{r} \phi \right ] + \frac{1}{r^n \sqrt{g_{_{S^n}} }} \partial _{\theta _i} \left [r^n \sqrt{g_{_{S^n}} } g^{\theta _i \theta _i}_{\mathbb{R}^{n+1}} \partial _{\theta _i} \phi \right ]$</code></p> <p><code>$=\frac{1}{r^n}\partial _r (r^n \partial_r \phi) + \frac{1}{\sqrt{g_{_{S^n}} }} \partial_{\theta _i} \left[ \sqrt{g_{_{S^n}} } \frac{g^{\theta _i \theta _i}_{S^n}} {r^2} \partial _{\theta _i} \phi \right]$</code> </p> <p>$=\frac{1}{r^n}\partial _r (r^n \partial_r \phi) + \frac{\nabla _{S^n}^2 \phi}{r^2}$ </p> <p>Therefore after doing the differentiation we have the final result,</p> <p>$$\nabla _{\mathbb{R}^{n+1}}^2 \phi = \frac{n}{r}\partial _r \phi + \partial _r ^2 \phi + \frac{\nabla _{S^n} ^2 \phi}{r^2}$$ </p> <p>And I don't see an neat way of writing the Laplacian on $S^n$ ! </p> http://mathoverflow.net/questions/10272/looking-for-a-reference-for-the-laplacian-operator/10601#10601 Answer by S. Carnahan for Looking for a reference for the laplacian operator S. Carnahan 2010-01-03T17:06:46Z 2010-01-03T17:06:46Z <p>I recommend Terras, <i>Harmonic analysis on symmetric spaces</i>.</p>