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 $u, v \in \map {C^\infty} M$ 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:
- $\ds \int_M u \nabla^2 v \rd V_g = \int_{\partial M} u N v \rd V_{\hat g} - \int_M \innerprod {\grad u} {\grad v}_g \rd V_g$
- $\ds \int_M \paren {u \nabla^2 v - v \nabla^2 u} \rd V_g = \int_{\partial M} \paren {u N v - v N u} \rd V_{\hat g}$
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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