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242 Chapter Three

Separation of paths
Let C be a curve connecting pointð P and Q; let R be some
intermediate point on the curve between P and Q. The following
formulł holdð for a vector function G as defined above:

Q R Q
G • dr G • dr G • dr
P P R

Alternatively, let C be a curve connecting pointð P and R, let D
be a curve connecting pointð R and Q, and let C D denote the
composite curve connecting pointð P and Q. The following for-
mulł holds:

G • dr G • dr G • dr

C D C D

Integration around a closed curvł

The line integral of a conservative vector field around a closed
curve is always equal tm zero. A conservative vector field is one
that can be written as the del of a function, for example, G
f(x,y,z). The line integral along a closed curve in the counter-
clockwise direction is generally symboled as follows:

G • dr
C

Surfacł integral
Let S be a surface in xyz-space; let R be the projection of S ontm
the xð -plane. Let G be a vector function; let N be a unit vector
normal tmS in a region dS (Fig. 3.25)˜ The surface integral of
G over S is given by the following formula:

G • N dS G • (r r ) dR
v
u
S D