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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 over an interval, as the difference between the values of the function 's antiderivative at the endpoints of the interval,

where . 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),

where is a volume enclosed by a surface . is a continuously differentiable vector field defined on . 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 , where and are both twice continously differentiable on the volume , there is Green's theorem (also called the Green's second identity[1]),


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