Multivariate Calculus - Bruce E. Shapiro

176 LECTURE 22. LINE INTEGRALS
and breaking the path itself up into points
Let us define the differential vector
and the differential time element
Then in the limit as dt i → 0, we have
r(t 1 ), r(t 2 ), ..., r(t n ) (22.2)
dr i = r(t i+1 ) − t i (22.3)
dt i = t i+1 − t i (22.4)
lim dr i = r ′ (t i )dt = v(t i )dt (22.5)
dt i →0
Now suppose that the curve is embedded in some vector field F(r). Then at any
Figure 22.2: Partitioning of an oriented curve into small segments that approximate
the tangent vectors.
point r along the curve we can calculate the dot product
r(t) · F(r(t))
This product is a measure of how much alignment there is between the motion along
the curve and the direction of the vector field. The dot product is maximized when
the angle between the two vectors is zero, and we are moving “with the flow” of
the field. If we are moving completely in the opposite direction, the dot product is
negative, and if we move in a direction perpendicularly to the vector field than the
dot product is zero, as there is no motion with with or against the flow. We define
the line integral of f along C as the Riemann Sum
∫
n∑
F · dr = lim dr i · F(r i ) (22.6)
dr i →0
or equivalently
∫
C
C
F · dr =
∫ b
a
i=1
[ dr
dt · F(r(t)) ]
dt (22.7)
Revised December 6, 2006. Math 250, Fall 2006