Exercise 1911Calculus › Multivariable calculus › Vector calculusM1Added 3 Nov 2026

Green's reciprocity

#Green#reciprocity#symmetry

Statement

State Green's reciprocity identity: for functions $u,v$ on a domain $D$ with boundary $\partial D$ ,
$\iint_D(u\Delta v-v\Delta u)=\oint_{\partial D}\left(u\frac{\partial v}{\partial n}-v\frac{\partial u}{\partial n}\right)ds.$

Hint

Apply the divergence theorem to $\vec F=u\nabla v-v\nabla u$ .

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Solution

Divergence. $\nabla\cdot(u\nabla v)=u\Delta v+\nabla u\cdot\nabla v$ . Similarly for $v\nabla u$ . Difference: $u\Delta v-v\Delta u$ .

Apply. Divergence theorem gives the boundary identity.

Conclusion. Green's reciprocity identity follows from the divergence theorem.

Green's reciprocity

Statement

Solution

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