Mathematics of Physics and Engineering

130 Functions of Several Variables

|| V/(Pfc)|| < 10 m for some positive integer m).

3.1.2 Integration and Differentiation

Let us recall the definition of the Riemann integral from the one-variable
calculus:

l f{x)dx = lim V/(4)Aa;fc,

J a maxAn-40^

where a = xo < • • • < XN = b is a partition of the interval [a,b\, Axk =
Xk — Xk-i, and x*k is a point in the interval [xk-i,Xk. By definition, the
function / is Riemann-integrable if the limit exists and does not depend on
the particular sequence of partitions or on the choice of the points x^. The
same definition is extendable to real-valued functions / defined on a set G
other than an interval, as long as there exists a function m that measures
the sizes of the subsets of the set G; we will not discuss the deep question
of measurability. Using the measure m, we define

r N

fdm= lim V/(P*)rn(Gfc), (3.1.13)

JG maxm(Gfc)-»0f^

where Gi,..., Gjv are mutually disjoint (Gk C\ Gm = 0) sets such that
their union |Jfc=1 Gfe contains the set G and every set Gk is measurable
(the size m(Gfc) of Gk is defined) and has a non-empty intersection with G
(Gfc f) G ^ 0); Pk is a point in Gk f] G. Note that definition (3.1.13) is not
tied to any coordinate system.
We will integrate functions defined on intervals, curves, planar regions,
surfaces, and solid regions:

Set

Interval

Curve

Planar region

Surface

Solid region

Measure m

Length

Arc length

Area

Surface Area

Volume

dm

dx

ds

dA

da

dV

The limit in (3.1.13) must be independent of the particular choice of
the sets Gfc and the points Pk, which, in general, puts certain restrictions
on both the function / (integrability) and the set G (measurability). If / is
integrable over a measurable set G, then the average value of / over G