Green's Identities

Theorem

Let $\struct {M, g}$ be a compact Riemannian manifold with or without boundary $\partial M$.

Let $\hat g$ be the induced metric on $\partial M$.

Let $\map {C^\infty} M$ be the smooth function space.

Let $u, v \in \map {C^\infty} M$ be smooth real functions on $M$.

Let $\rd V_g$ be the Riemannian volume form.

Let $\nabla^2$ be the Laplace-Beltrami operator.

Let $\grad$ be the gradient operator.

Let $N$ be the outward-pointing unit normal vector field on $\partial M$.

Let $\innerprod \cdot \cdot_g$ be the Riemannian metric with respect to the metric $g$.

Then:

$\ds \int_M u \nabla^2 v \rd V_g = \int_{\partial M} u N v \rd V_{\hat g} - \int_M \innerprod {\grad u} {\grad v}_g \rd V_g$

$\ds \int_M \paren {u \nabla^2 v - v \nabla^2 u} \rd V_g = \int_{\partial M} \paren {u N v - v N u} \rd V_{\hat g}$

Further research is required in order to fill out the details. In particular: There are also $\R^3$ identities, often presented separately, and there are 3 of them, the third being Laplacian of Green function You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by finding out more. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Research}} from the code.

Proof

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Source of Name

This entry was named for George Green.

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Problems