Later, in 1907, it was extended by Levi-Civita to flows separating from a smooth curved boundary.

As it is easily seen, this is a version of the divergence theorem for domains with non smooth boundary.

This method can be applied to a shape with an irregular, smooth or complex boundary where other methods are too difficult.

For smooth boundary, this was proved by Victor Ivrii in 1980.

Consider a bounded open set with smooth boundary , and a continuous function .

Let Ω be a bounded domain in R with smooth boundary.

The theory for multiply connected bounded domains with smooth boundary follows easily from the simply connected case.

Let Ω be a bounded domain in n-dimensional Euclidean space with smooth boundary.

A smooth boundary is evidently two-sided, but a non-smooth (especially, fractal) boundary can be quite complicated.

This situation occurs, for instance, when p lies on a thin plate-like feature, or on the smooth boundary between two regions with contrasting values.