The Laplace Equation and Wave Equation

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


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


Vector Field Wave Equation Riemannian Manifold Covariant Derivative Laplace Operator 
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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of North CarolinaChapel HillUSA

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