Begin2.DVI

s 0 < s 1 <... < sn,

where corresponding to each value of the arc length parameter sithere is a position

vector
r (si) = x(si)ˆe 1 +y(si)ˆe 2 +z(si)ˆe 3 , for i= 1,... , n , as illustrated in figure 6-18.

A change in the element of arc length from
r (si− 1 )to
r (si)is defined as

∆si=|
r (si)−
r (si− 1 )|=|∆
r i|.

The total arc length is obtained from the sum of these elements of arc length as

the number of these lengths increase without bound and the partition gets finer and

finer. In symbols, this limit is denoted as

s= limn→∞

∑n

i=1

∆si=

∫sn

s 0

ds.

The above definition for arc length along the curve suggests how values of a scalar

field can be summed as one moves through the scalar field along a curve C.

Definition (Line integral of a scalar function along a curve C.)

Let f = f(x, y, z ) denote a scalar function of position. The line

integral of f along a curve Cis defined as

∫

C

f(x, y, z )ds = limn→∞

∑n

i=1

f(x∗i, y ∗i, z ∗i) ∆ si, (6 .94)

where (x∗i, y i∗, z ∗i)is a point on the curve in the ith subinterval ∆si

and where the symbol

∫

C denotes an integral taken along the given

curve C. This type of integral is called a line integral along the

curve.

Similarly, define the summation of a vector field as one moves through the field

along a curve C. This produces the following definition of a line integral of a vector

function along a curve C.

Definition (Line integral along a curve Cinvolving a dot product.) Let

F=
F(x, y, z ) = F 1 (x, y, z ) ˆe 1 +F 2 (x, y, z ) ˆe 2 +F 3 (x, y, z ) ˆe 3

denote a vector function of position. The line integral of
F along a given curve C,

defined by a position vector
r =
r (s) = x(s)ˆe 1 +y(s)ˆe 2 +z(s) ˆe 3 ,is defined as