Green's Identities

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

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

Let $\Delta$ be the Laplacian 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:

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



Proof




Source of Name

This entry was named for George Green.


Sources