Differential Operators on Semi‐Riemannian Manifolds
Stephen C. Newman · 2019
This chapter presents a discussing of differential operators on semi-Riemannian manifolds, including Hodge star, codifferential operators, gradient operators, divergence of vector fields, curl operators, Hesse operator, Laplace operator, and Laplace–de Rham operator. It also includes the discussion of divergence of symmetric 2-covariant tensor fields. The chapter explains that the Hodge star, codifferential, and the Laplace–de Rham operator are the family of linear maps. It shows that the Curl and classical curl operators are identical in R03. The chapter also shows that, except for a sign, the Laplace–de Rham operator is a generalization of the Laplace operator. It provides many worked examples and detailed diagrams to aid understanding. The chapter appeals especially to physics students wishing to learn more differential geometry than is usually provided in texts on general relativity.