# User:Zhennan/Greenstheorem

Green's theorem can find its root in the fundamental theorem of integral calculus. The fundamental theorem of integral calculus expresses the value of a definite integral of a given integrable function ${\displaystyle f}$ over an interval, as the difference between the values of the function ${\displaystyle f}$'s antiderivative ${\displaystyle F}$ at the endpoints of the interval,

${\displaystyle \int _{a}^{b}f(x)dx=F(b)-F(a)}$

where ${\displaystyle F'(x)=f(x)}$. This is a fundamental tool to solve problems within a restricted region or interval. The multidimensional extension of this theorem is divergence theorem (also called Gauss's theorem),

${\displaystyle \int _{V}(\nabla \cdot {\textbf {A}})d{\textbf {r}}'=\oint _{S}{\textbf {A}}\cdot {\hat {\textbf {n}}}\,dS}$

where ${\displaystyle V}$ is a volume enclosed by a surface ${\displaystyle S}$. ${\displaystyle {\textbf {A}}}$ is a continuously differentiable vector field defined on ${\displaystyle V}$. Physically, the divergence theorem relates the normal outflow of a vector field through a closed surface to the volume integration of the divergence of that field. Choosing ${\displaystyle {\textbf {A}}=\phi \nabla \psi -\psi \nabla \phi }$, where ${\displaystyle \phi }$ and ${\displaystyle \psi }$ are both twice continously differentiable on the volume ${\displaystyle V}$, there is Green's theorem (also called the Green's second identity[1]),

${\displaystyle \int _{V}(\phi \nabla ^{2}\psi -\psi \nabla ^{2}\phi )d{\textbf {r}}=\oint _{S}(\phi \nabla \psi -\psi \nabla \phi )\cdot {\hat {\textbf {n}}}\,dS}$

## References

1. [1] Aki, K., & Richards, P. G. (2002). Quantitative seismology.