Mathematics of Physics and Engineering

Integration and Differentiation 131

is, by definition,

lG=

w)L

fdm

(3

Li4)

The idea of the Riemann integral was introduce by the German mathemati-
cian GEORG FRIEDRICH BERNHARD RIEMANN (1826-1866) in 1854 in his
Habilitation dissertation (Habilitation is one step higher than Ph.D. and
is the terminal degree in several European countries).

We now summarize the main facts about line, area, surface, and volume
integrals.
LINE, OR PATH, INTEGRALS. The set G in (3.1.13) is a curve C defined
by the vector-valued function r = r(t), a < t < b. The set is measurable
and has the line element ds if C is piece-wise smooth; see pages 28-29.
The line integral of the first kind Jcfds is defined for a continu-
ous scalar field / by the equality

[ fds= [ f(r(t))\\r'(t)\\dt, (3.1.15)

JC Ja

where f{r(t)) is the value of the scalar field / at the point on the curve
with position vector r{t). If f{P) represents the linear density of the curve
at the point P, then Jc f ds is the mass of the curve.
The line integral of the second kind LF-dr is defined for a
continuous vector field F by the equality

f Fdr= f F(r{t))-r'{t)dt, (3.1.16)

JC Ja

where F(r(t)) is the value of the vector field F at the point on the curve
with position vector r(t). For a simple closed curve C, the corresponding
line integral of the second kind is denoted by §CF • dr and is called the
circulation of F along C. The line integral of the second kind can be
reduced to the line integral of the first kind:

L'-*-L

F-uds, (3.1.17)

where u is the unit tangent vector to the curve. The direction of u defines
the orientation of C; the integral reverses sign if the orientation of C is
reversed. In cartesian coordinates, F = Fi i + F2 j + F 3 K, and JCF -dr is
often written as jc F\dx + i^dy + F$dz.