Divergence theorem

In 1826, Ostrogradsky gave the first general proof of the divergence theorem, which was discovered by Lagrange in 1762.^[10] This theorem may be expressed using Ostrogradsky's equation:

${\displaystyle \iiint _{V}\left({\partial P \over \partial x}+{\partial Q \over \partial y}+{\partial R \over \partial z}\right)dx\,dy\,dz=\iint _{\Sigma }\left(P\cos \lambda +Q\cos \mu +R\cos \nu \right)d\Sigma }$;

where P, Q, and R are differentiable functions of x, y, and z defined on the compact region V bounded by a smooth closed surface Σ; λ, μ, and ν are the angles that the outward normal to Σ makes with the positive x, y, and z axes respectively; and dΣ is the surface area element on Σ.

Ostrogradsky's integration method

His method for integrating rational functions^[11] is well known. First, we separate the rational part of the integral of a fractional rational function, the sum of the rational part (algebraic fraction) and the transcendental part (with the logarithm and the arctangent). Second, we determine the rational part without integrating it, and we assign a given integral in Ostrogradsky's form:

${\displaystyle \int {R(x) \over P(x)}\,dx={T(x) \over S(x)}+\int {X(x) \over Y(x)}\,dx,}$

where ${\displaystyle P(x),\,S(x),\,Y(x)}$ are known polynomials of degrees p, s, y respectively; ${\displaystyle R(x)}$ is a known polynomial of degree not greater than ${\displaystyle p-1}$; and ${\displaystyle T(x),\,X(x)}$ are unknown polynomials of degrees not greater than ${\displaystyle s-1}$ and ${\displaystyle y-1}$ respectively.

Third, ${\displaystyle S(x)}$ is the greatest common divisor of ${\displaystyle P(x)}$ and ${\displaystyle P'(x)}$. Fourth, the denominator of the remaining integral ${\displaystyle Y(x)}$ can be calculated from the equation ${\displaystyle P(x)=S(x)\,Y(x)}$.

When we differentiate both sides of the equation above, we get:
${\displaystyle R(x)=T'(x)Y(x)-T(x)H(x)+X(x)S(x)}$,

where ${\displaystyle H(x)={Y(x)S'(x) \over S(x)}}$.

It can be shown that ${\displaystyle H(x)}$ is polynomial.