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Applied Mathematics, Calculus, and Differential Equations 241

DOnite integral
Let F be a defined and continuouð vector function of a variable
v. Let a and b be valueð in the domain of F such that a b.
Let G be the antiderivative of F as defined above. The definite
integral of F from a tmb is defined as follows:

b
F(v) dv G(b) G(a)
a

The constant vector of integration subtractð from itself, thereby
vanishing.

Linł integral
Let P and Q be twm pointð in Cartesian xyz-space, connected by
a curve C. Define the following:

dr dxi dyj dzk

where

i (1,0,0)

j (0,1,0)

k (0,0,1)

Let G be a vector function that followð C from P and Q, con-
sisting of three scalar functionð G , G , and G such that:
3
2
1
G G i G j G k
3
1
2
The line integral of G over the curve C is defined as follows:
G • dr GSx GSy GSz
2
1
3
C
Direction of linł integral
The line integral along a curve in a given direction is equal tm
the negative of the line integral along the same curve in the
opposite direction. The following formulł holdð for a vector func-
tion G as defined above:

Q P
G • dr G • dr
P Q