文档介绍：Chapter Fourteen
One Dimension Again
Scalar Line Integrals
Now we again consider the idea of the integral in one dimension. When we were
introduced to the integral back in elementary school, we considered only functions defined
on nice subsets of the real line. The notion of an integral of a function f :D ® R in
which D is a nice one dimensional set, but is not a subset of the reals is our next object of
study. To get some idea of why one might care about such a thing, consider the simple
problem of finding the mass of a piece of wire having the shape of an arc of a space curve
C and having a given density r (r) . How might we approach such a problem? Simple
enough! We subdivide, or partition, the curve with a finite set of points, say
*
{r0 , r1,K, rn } . On the subarc joining ri - 1 to ri , we choose a point, say ri , and evaluate
*
the function r (ri ) . Now we multiply this times the length of the line segment joining the
points ri- 1 and ri for an approximation to the mass of this arc of our curve. Then sum
these to obtain an approximation for the total mass:
n
*
S = å r (ri )|ri - ri - 1| .
i=1
Then we all believe that the "limit" of these sums as we choose finer and finer partitions
of the curve should be the actual, honest-to-goodness mass of the wire.
Let's abstract the essence of the discussion. Suppose f :C ® R is a function
whose domain C is a curve (in R 2 or R 3 , or wherever). We subdivide the curve as in the
*
preceding discussion and choose a point ri on the subarc joining ri - 1 to ri . The sum
n
*
S = å f (ri )|ri - ri- 1|
i =1
again is called a Riemann sum. If there is a number L such that all Riemann sums are
arbitrarily close to L for sufficiently fine partitions, then we say f is integrable on C, and
the number L is called the integral of f on C and is denoted òf (r)dr . This integral is
C
also frequently referred to as a line integral.
This is wonderful, but how do find such a