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Numerical calculation method for function integration on submanifolds of $\mathbb{R}^n$ or compact Riemannian manifolds

Fusheng Deng, Gang Huang, Yingyi Wu

Abstract

In this paper, we present a method for digitally representing the "volume element" and calculating the integral of a function on compact hypersurfaces with or without boundary, and low-dimensional submanifolds in $\mathbb{R}^n$. We also extend such calculation to hypersurfaces in compact Riemannnian manifolds.

Numerical calculation method for function integration on submanifolds of $\mathbb{R}^n$ or compact Riemannian manifolds

Abstract

In this paper, we present a method for digitally representing the "volume element" and calculating the integral of a function on compact hypersurfaces with or without boundary, and low-dimensional submanifolds in . We also extend such calculation to hypersurfaces in compact Riemannnian manifolds.
Paper Structure (10 sections, 7 theorems, 70 equations, 2 figures)

This paper contains 10 sections, 7 theorems, 70 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The divergence theorem
  4. Riemannian coordinates
  5. Two more lemmas
  6. Numerical calculation method for function integration in submanifold in $\mathbb{R}^n$
  7. Hypersurfaces in $\mathbb{R}^n$
  8. Hypersurfaces with Boundaries in $\mathbb{R}^n$
  9. Low-dimensional submanifold in $\mathbb{R}^n$
  10. Numerical calculation method for function integration in Riemannian manifolds

Key Result

Theorem 2.1

(Lee00 Theorem 16.32) (The Divergence Theorem) Let $(M, g)$ be an oriented Riemannian manifold with boundary. For any compactly supported smooth vector field $X$ on $M$, where $\vec{n}$ is the outward-pointing unit normal vector field along $\partial M$, $dV_{M}$ is the Riemannian volume form of $M$ and $dV_{\partial M}$ is the Riemannian volume form of $\partial M$ induced from $g$.

Figures (2)

  • Figure 1: hypersurface with boundaries
  • Figure 2: Low dimensional submanifold

Theorems & Definitions (8)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 3.1: F95
  • Theorem 4.1
  • proof