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class="docs-label">Basic manifolds</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../manifolds/centeredmatrices.html">Centered matrices</a></li><li><a class="tocitem" href="../manifolds/choleskyspace.html">Cholesky space</a></li><li><a class="tocitem" href="../manifolds/circle.html">Circle</a></li><li><a class="tocitem" href="../manifolds/elliptope.html">Elliptope</a></li><li><a class="tocitem" href="../manifolds/essentialmanifold.html">Essential manifold</a></li><li><a class="tocitem" href="../manifolds/euclidean.html">Euclidean</a></li><li><a class="tocitem" href="../manifolds/fixedrankmatrices.html">Fixed-rank matrices</a></li><li><a class="tocitem" href="../manifolds/flag.html">Flag</a></li><li><a class="tocitem" href="../manifolds/generalizedstiefel.html">Generalized Stiefel</a></li><li><a class="tocitem" href="../manifolds/generalizedgrassmann.html">Generalized Grassmann</a></li><li><a class="tocitem" 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for="menuitem-3-3"><span class="docs-label">Manifold decorators</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../manifolds/connection.html">Connection manifold</a></li><li><a class="tocitem" href="../manifolds/group.html">Group manifold</a></li><li><a class="tocitem" href="../manifolds/metric.html">Metric manifold</a></li><li><a class="tocitem" href="../manifolds/quotient.html">Quotient manifold</a></li></ul></li></ul></li><li><span class="tocitem">Features on Manifolds</span><ul><li><a class="tocitem" href="atlases.html">Atlases and charts</a></li><li><a class="tocitem" href="differentiation.html">Differentiation</a></li><li><a class="tocitem" href="distributions.html">Distributions</a></li><li class="is-active"><a class="tocitem" href="integration.html">Integration</a></li><li><a class="tocitem" href="statistics.html">Statistics</a></li><li><a class="tocitem" href="testing.html">Testing</a></li><li><a class="tocitem" 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href="integration.html">Integration</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href="integration.html">Integration</a></li></ul></nav><div class="docs-right"><a class="docs-edit-link" href="https://github.com/JuliaManifolds/Manifolds.jl/blob/master/docs/src/features/integration.md" title="Edit on GitHub"><span class="docs-icon fab"></span><span class="docs-label is-hidden-touch">Edit on GitHub</span></a><a class="docs-settings-button fas fa-cog" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-sidebar-button fa fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a></div></header><article class="content" id="documenter-page"><h1 id="Integration"><a class="docs-heading-anchor" href="#Integration">Integration</a><a id="Integration-1"></a><a class="docs-heading-anchor-permalink" href="#Integration" title="Permalink"></a></h1><article class="docstring"><header><a class="docstring-binding" id="Manifolds.manifold_volume-Tuple{AbstractManifold}" href="#Manifolds.manifold_volume-Tuple{AbstractManifold}"><code>Manifolds.manifold_volume</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia hljs">manifold_volume(M::AbstractManifold)</code></pre><p>Volume of manifold <code>M</code> defined through integration of Riemannian volume element in a chart. Note that for many manifolds there is no universal agreement over the exact ranges over which the integration should happen. For details see <a href="../misc/references.html#BoyaSudarshanTilma:2003">[BST03]</a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/JuliaManifolds/Manifolds.jl/blob/fb62af238628612ae7c13e48cc9b862125e5d59c/src/Manifolds.jl#L495-L501">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="Manifolds.volume_density-Tuple{AbstractManifold, Any, Any}" href="#Manifolds.volume_density-Tuple{AbstractManifold, Any, Any}"><code>Manifolds.volume_density</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia hljs">volume_density(M::AbstractManifold, p, X)</code></pre><p>Volume density function of manifold <code>M</code>, i.e. determinant of the differential of exponential map <code>exp(M, p, X)</code>. Determinant can be understood as computed in a basis, from the matrix of the linear operator said differential corresponds to. Details are available in Section 4.1 of <sup class="footnote-reference"><a id="citeref-ChevallierLiLuDunson2022" href="#footnote-ChevallierLiLuDunson2022">[ChevallierLiLuDunson2022]</a></sup>.</p><p>Note that volume density is well-defined only for <code>X</code> for which <code>exp(M, p, X)</code> is injective.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/JuliaManifolds/Manifolds.jl/blob/fb62af238628612ae7c13e48cc9b862125e5d59c/src/Manifolds.jl#L504-L518">source</a></section></article><section class="footnotes is-size-7"><ul><li class="footnote" id="footnote-ChevallierLiLuDunson2022"><a class="tag is-link" href="#citeref-ChevallierLiLuDunson2022">ChevallierLiLuDunson2022</a><blockquote><p>E. Chevallier, D. Li, Y. Lu, and D. B. Dunson, “Exponential-wrapped distributions on symmetric spaces.” arXiv, Oct. 09, 2022. doi: <a href="https://doi.org/10.48550/arXiv.2009.01983">10.48550/arXiv.2009.01983</a>.</p></blockquote></li></ul></section></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="distributions.html">« Distributions</a><a class="docs-footer-nextpage" href="statistics.html">Statistics »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Wednesday 30 August 2023 12:58">Wednesday 30 August 2023</span>. 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