Calculus: Analytic Geometry and Calculus, with Vectors

13.8 Triple integrals; spherical coordinates 707

where G is the, gravitational constant and the last factor isa unit vector in the
direction of QPk Making a slight modification of the right member of (1) and
adding give the right member of the formula

(2) RS = Gm I 8(Px)QPx

ASk,

k1 IQPxl3

Which is a Riemann sum (RS) formed for the vector function having the value

(3) a (P) QP

l0i,

at the point P of our set S. When we are in a hurry, we use the idea that the

Riemann sum should be near the force F whenei er the norm of the partition is

small as a basis for introducing the definition

(4) F = lim Cm ° S(Pk)QPxpSx.

k=1 lQPkl3

Since the limit of the Riemann sums is a Riemann integral, we write

(5) F = Gm f

S(P)QP

dS,

s 1013

the symbol on the right being an orthodox symbol for the integral. When we

wish our notation to be as informative as possible, we can use n integral signs

when S is n-dimensional. The fact that different notations are used at different

times need not disturb us, because in any particular application we can be

expected to know the dimensionality of the set we partition. We can, when we

are unhurried, be more precise about the meanings of (4) and (5). The integral

is, when it exists, the one and only vector F such that to each positive number e

there corresponds a positive number S such that

(6) F - Gm i S(PI)QPAS, <

kI lQPkl3

whenever the Riemann sum is formed for a partition whose norm is less than S.

For some purposes, it is important to observe that (5) is an intrinsic formula

which does not depend upon any one particular coordinate system which may

be used to specify the positions of the points involved in the problem. For

other purposes, particularly when problems in elementary books are being solved,

it is necessary to introduce a coordinate system. The raison d'etre of different

coordinate systems lies in the fact that different ones are most useful in different

situations.

13.8 Triple integrals; spherical coordinates In some cases a solid S
(or a set S in Es having positive volume) and a function f defined over S
are such that the triple integral defined by the formula

(13.81) f f fsf(P)dS=limI f(P)AS