Begin2.DVI

represents a scalar which is the sum of the projections of F
onto the normals to the

surface elements. If the surface is divided into nsmall surface elements ∆Si,where

i= 1 ,... , n. Let F
i=F
(xi, yi, zi)represent the value of the vector field over the ith

surface element. The summation of the elements

F
i·∆S
i=F
i·ˆeni∆Si

over all surface elements represents the sum of the normal components of F
imulti-

plied by ∆Sias ivaries from 1 to n. A summation gives the surface integral

nlim→∞

∑n

i=1

F
i·∆S
i=

∫∫

R

F
·dS .
 (7 .99)

Again, the form of this integral depends upon how the given surface is represented.

Integrals of this type arise when calculating the volume rate of change associated

with velocity fields. It is called a flux integral and represents the amount of a

substance moving across an imaginary surface placed within the vector field.

The vector integral ∫∫

R

F
×dS

represents a vector which is obtained by summing the vector elements F
i×∆S
iover

the given surface. The fundamental theorem of integral calculus enables such sums

to be expressed as integrals and one can write

nlim→∞

∑n

i=1

F
i×∆S
i=

∫∫

R

F
×dS .
 (7 .100)

Integrals of this type arise as special cases of some integral theorems that are devel-

oped in the next chapter.

Each of the above surface integrals can be represented in different forms de-

pending upon how the element of surface area is represented. The form in which

the given surface is represented usually dictates the method used to calculate the

surface area element. Sometimes the representation of a surface in a different form

is helpful in determining the limits of integration to certain surface integrals.