# 48. Three equivalent definitions of the Laplacian

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* - The
*gradient* - The
*Hessian* - The
*Laplacian*

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

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

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

Comments and questions?