Mathematical Physics Vol 1

Chapter 2. Vector analysis

54

where α , β and γ are angles the vector n forms with the coordinate axes Ox , Oy and Oz , respectively. The integral can now be represented in the form x S ϕ ( X ) · d S = x S v · d S = x S v · n d S = x S ( v x cos α + v y cos β + v z cos γ ) d S . (2.78) On the other hand, let the equation of the surface S be

z = f ( x , y ) ,

(2.79)

then

· ± q 1 + f ′ 2

′ 2 y d x d y =

x S

v z d x d y = x D xy

v z ± 1 q 1 + f ′ 2 v z cos γ d S ,

x + f

′ 2

x + f

y

= x S

(2.80)

where D xy is the projection of the surface S on the plane Oxy , and γ the angle between the normal n and the z –axis. The sign ± depends on whether the integration is performed on the upper or the lower part of the surface. A similar result is obtained for the other two integrals x S v y d z d x = x S v y cos β d S , (2.81) x S v x d y d z = x S v x cos α d S . (2.82) By substituting the relations (2.80)-(2.82) in (2.78), we obtain x S v x d y d z + v y d z d x + v z d x d y = x S ( v x cos α + v y cos β + v z cos γ ) d S , the relation between the surface integral with respect to coordinates and surface integral with respect to surface.