The divergence theorem, more commonly known especially in older literature as Gauss's theorem (e.g., Arfken 1985) and also known as the Gauss-Ostrogradsky theorem, is a theorem in vector calculus that can be stated as follows. Let be a region in space with boundary . Then the volume integral of the divergence of over and the surface integral of over the boundary of are related by
(1)
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The divergence theorem is a mathematical statement of the physical fact that, in the absence of the creation or destruction of matter, the density within a region of space can change only by having it flow into or away from the region through its boundary.
A special case of the divergence theorem follows by specializing to the plane. Letting be a region in the plane with boundary , equation (1) then collapses to
(2)
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If the vector field satisfies certain constraints, simplified forms can be used. For example, if where is a constant vector , then
(3)
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But
(4)
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so
(5)
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(6)
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and
(7)
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But , and must vary with so that cannot always equal zero. Therefore,
(8)
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Similarly, if , where is a constant vector , then
(9)
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