Mathematical Physics Vol 1

2.2 Integration

53

Vector surface integral Let ϕ ( r )= ϕ ( X )= ϕ ( x , y , z ) be a continuous scalar or vector function of the point X ( x , y , z ) , that is, of the position vector r of this point. Let us then partition the surface S into a finite number of parts ∆ S i , i = 1 , 2 ,..., n . The area of each part ∆ S i can be represented in vector form, that is, as a vector whose intensity is equal to the area of ∆ S i ∆ S i = ∆ S i n , (2.73) where n is the unit vector of the surface normal at an arbitrary point X i which belongs to ∆ S i .

Figure 2.11: Part of the surface ∆ S i .

Let us now form the integral sum

n ∑ i = 1

ϕ ( X i ) ◦ ∆ S i ,

I =

(2.74)

where X i ∈ ∆ S i . This sum can be a scalar or vector variable, depending on the nature of the function ϕ and the meaning of the ◦ symbol for the product (scalar or vector product). Definition The limit value of the sum I , when the largest absolute value | ∆ S i | tends to zero, is called the vector surface integral of the function ϕ over the surface S and denoted by

ϕ ( X i ) ◦ ∆ S i = x S

n ∑ i = 1

ϕ ( X ) ◦ d S ,

lim n → ∞

(2.75)

if such a limit value exists.

Analysis Let ϕ be a vector function, represented in one of the following ways ϕ ( X ) ≡ v = v x i + v y j + v z k =[ v x , v y , v z ] , (2.76) and let the ◦ symbol stand for the scalar product. The unit vector of the normal can be represented in the form n = cos α i + cos β j + cos γ k =[ cos α , cos β , cos γ ] , (2.77)