Divergence
Definition

The divergence of a vector field F (x) at a point x is defined as the limit of the ratio of the surface integral of F out of the closed surface of a volume V enclosing x to the volume of V, as V shrinks to zero

where | V | is the volume of V, S (V) is the boundary of V, and n ^ {\displaystyle \mathbf {\hat {n}} } is the outward unit normal to that surface. It can be shown that the above limit always converges to the same value for any sequence of volumes that contain x and approach zero volume. The result, div F, is a scalar function of x.

Since this definition is coordinate-free, it shows that the divergence is the same in any coordinate system. However it is not often used practically to calculate divergence; when the vector field is given in a coordinate system the coordinate definitions below are much simpler to use.

A vector field with zero divergence everywhere is called solenoidal – in which case any closed surface has no net flux across it.
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