Calculus: Analytic Geometry and Calculus, with Vectors

4.6 Riemann-Cauchy integrals and work

It follows from theseformulas that

(4.673) lim Wa,b =

r°° k=1 dx

b.. X a

25S

This formula is responsible for some terminology that scientists often use.

The limit in (4.673) is called "the amount of work required to take the

test particle from a toinfinity" and this amount of work is called the

potential (or gravitational potential) at the point a due to the particle of

mass ml at the origin. It is an easy consequence of these definitions and

formulas that the potential, say u, at the point P(x,y,z) due to a particle

of mass ml concentrated at the point Po(xo,yo,zo) is

(4.68) u=

km1

'f (x -xo)2+ (y- yo)2+ (z- zo)2

The basic importance of the concept of potential u lies in the fact that if a
particle of mass m is moved from a point P1 to a point P2 with no forces
upon it except gravitational forces and a force F, and if the speeds at
P1 and P2 are equal, then the work done by the force F is equal to the
product of m and the potential difference, that is, the potential at the
starting point Pi minus the potential at the destination P2.
All of the above ideas and formulas apply to electrostatic potentials as
well as to gravitational potentials. In the electrical case, we start with

two charges qi and q2 and apply the Coulomb (1736-1806) law

IFl = kgig2/x2, which is the electrical analogue of the Newton law of

gravitation.

Problems 4.69

1 Suppose somebody writes

I l1dx= oo,

o x f 1dx= oo

1 X

with the hope that he is conveying information to you.

Ans.:

What does he mean?

lim 1 1 dx= oo, lim I 1 dx = oo.

h->o+ h x h-'.0 1 x

2 Prove that

i

r dx = o0

Ii'XPdx P-1' JoXP

dx = oo,

w 1 1

11 xP 1o x' 1-P

3 Show that, when k > 0,

1,

e k: dx = k

0