Mathematical Physics Vol 1

Chapter 4. Field theory

102

4.2.8 Integral theorems In this part we will outline several theorems (Stokes’ 11 , Green’s 12 , Gauss’s 13 ) which are very often used in integral calculus and its applications. 14 Stokes’s theorem If the projections v x ( x , y , z ) , v y ( x , y , z ) and v z ( x , y , z ) , of a vector function v ( r ) are continuous, as well as their corresponding partial derivatives, on the surface S , which is closed by the spatial curve C , then I C v d r = x S ( ∇ × v ) · n d S = x S ( ∇ × v ) · d S = x S rot v · d S ! , (4.90) where n is the unit vector of the normal to the observed surface. Green’s theorem If for a scalar function Φ there exists a line integral along a closed line C , and if grad Φ is a continuous function in the region S bordered by the curve C , then I C Φ d r = x S ( n × ∇Φ ) d S = x S d S × ∇Φ ! . (4.91) Gauss’s theorem If for a vector function v ( r ) there exists a surface integral along a closed surface S , which represents the border of a region V and if div v is a continuous function in this region, then y V ∇ · v d V = x S v · n d S = x S v · d S ! . (4.92) This theorem is also known as the divergence theorem or the Gauss–Ostrogradsky theorem. 15 . The mean value theorem 1. If f ( x , y ) is a continuous function on a closed and limited region σ in the x , y plane, then there exists at least one point ( x o , y o ) ∈ σ such that s σ f ( x , y ) d σ = f ( x o , y o ) · P , where P is the area of region σ . 2. If f ( x , y , z ) is a continuous function on a closed and limited region σ in space, then there exists at least one point ( x o , y o , z o ) ∈ σ such that y σ f ( x , y , z ) d σ = f ( x o , y o , z o ) · V , (4.93) where V is the volume of region σ . 11 Stokes, George Gabriel (1819–1903), Irish mathematician and physicist. He is known for his contributions to the theory of infinite series as well as contributions to fluid mechanics (Navier-Stokes equations), geodesy and optics. 12 Green, George (1793-1841), English mathematician. His work pertains to the theory of potentials related to electricity and magnetism, as well as to oscillations, waves and the theory of elasticity. 13 Gauss, Carl Friedrich (1777–1855), a great German mathematician. His work is of fundamental importance for algebra, number theory, differential equations, differential geometry, non-Euclidean geometry, complex analysis, astronomy, geodesy, electromagnetism and theoretical mechanics. 14 Proof of these theorems, due to limited space, is not given, but they are outlined here due to their importance. 15 Ostrogradskii$, Mihail Vasil mekieviq (1801-1862). Famous Russian mathematician and mechanics scientist.