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Under suitable conditions of
continuity and the existence of various partial derivatives, the double
integral for the area of a closed region can be re-stated as a line
integral around the region. Green's Theorem is derived from a more general
theorem in multivariable calculus known as Stoke's Theorem. Essentially
this mean that it is possible, in some instances, to determine the value
of an integral across a closed region by considering only the boundary,
and disregarding the interior altogether.
George Green first established
his famous theorem around 1825.
Green's theorem allows for
some difficult integrals in multivariable calculus to be re-expressed
in another form. Occasionally this new form will be solvable, whereas
the original integral was not.
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