A review: The Laplacian and the d’Alembertian
A review: The Laplacian and the d’Alembertian
This chapter reviews the Laplacian and the d'Alembertian. It begins with a brief discussion on the solution of wave equation both in Euclidean space and on manifolds and how this knowledge can be used to derive properties of eigenfunctions on Riemannian manifolds. A key step in understanding properties of solutions of wave equations on manifolds is to compute the types of distributions that include the fundamental solution of the wave operator in Minkowski space (d'Alembertian), with a specific function for the Euclidean Laplacian on Rn. The chapter also reviews another equation involving the Laplacian, before discussing the fundamental solutions of the d'Alembertian in R1+n.
Keywords: Laplacian, d'Alembertian, wave equations, Euclidean space, Riemannian manifolds, eigenfunctions, Minkowski space, Euclidean Laplacian
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