Green's Formulas

formulas of integral calculus interrelating various types of integrals. The simplest of them relates a double integral over region G to a curvilinear integral along boundary C of region G and has the form

This formula was known even to L. Euler (1771). Two other formulas first published by George Green in 1828 in connection with research on the theory of potential are

(the first Green formula or the preliminary Green formula) and

Here, G is the region of a three-dimensional space, surface 5 is the boundary of this region, Δu = ∂²u/∂x² + ∂²u/∂y² + ∂²u/∂z² (Δv is defined in the same way) is a Laplace operator, and ∂u/∂n, ∂v ∂n are derivatives in the direction of the outer normal to S.