# What is the Divergence Theorem? Explain how it generalizes Green’s Theorem to three dimensions.

What is the Divergence Theorem? Explain how it generalizes Green’s Theorem to three dimensions.
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Nichole Watt
Divergence Theorem:
- Consider F be a vector field whose components consists of continuous first partial derivatives and considers be a piecewise smooth oriented closed surface.
- The flux of F across S in the direction of the surface’s outward unit normal field n equals the triple integral of the divergence grad*F over the solid region D bounded by the surface:
$\int {\int }_{S}F\cdot nd\sigma =\int {\int }_{D}\int \mathrm{\nabla }\cdot FdV$
Step 2
- The divergence theorem simplifies the normal (flux) form of Green’s
theorem in a two-dimensional region in the plane and a three-dimensional region in space.
The total flux of the field across the boundary bounding the region is equal to the integral of grad*F over the interior of the region.