Let $M$ be an oriented Riemannian manifold. In this lecture we make contact with some of the differential operators you first met in Analysis II:

• The divergence $\operatorname{div}(X)$ of a vector field,
• The gradient $\operatorname{grad}(f)$ of a smooth function,
• The Hessian $\operatorname{Hess ^{ \nabla}}(f)$ of a smooth function,
• The Laplacian $\Delta(f)$ of a smooth function.

But why define something once when you can define it three times? 🙃

For the sake of completeness we give three equivalent definitions of the Laplacian $\Delta(f)$:

1. $\Delta(f) = \operatorname{div}( \operatorname{grad}(f))$,
2. $\Delta(f) = - \delta(df)$,
3. $\Delta(f) = \operatorname{tr}( \operatorname{Hess}^{ \nabla}(f))$.