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The Laplace Equation and Wave Equation

  • Michael E. Taylor
Chapter
Part of the Applied Mathematical Sciences book series (AMS, volume 115)

Abstract

In this chapter we introduce the central linear partial differential equations of the second order, the Laplace equation
$$\Delta u = f$$
(0.1)
and the wave equation
$$\left (\frac{{\partial }^{2}} {\partial {t}^{2}} -\Delta\right )u = f.$$
(0.2)
For flat Euclidean space \({\mathcal{R}}^{n}\), the Laplace operator is defined by
$$\Delta u =\frac{{\partial }^{2}u} {\partial {x}_{1}^{2}} +\cdots +\frac{{\partial }^{2}u} {\partial {x}_{n}^{2}}.$$
(0.3)

Keywords

Vector Field Wave Equation Riemannian Manifold Covariant Derivative Laplace Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of North CarolinaChapel HillUSA

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