400 Poisson distribution

Polar coordinates

√12+ 12

the theory of gravitation in the years that followed. He

found applications of this work to the theory of elec-

tromagnetism in 1813 and to the theory of heat trans-

fer in 1815.

Late in his career, Poisson’s interests turned toward

probability. Fascinated by studies of societal behavior,

he began to analyze the probability of a random event

occurring within a given interval of time given that its

likelihood of happening is very small, but the number

of opportunities for it to occur is large. This work led

to his discovery of the P

OISSON DISTRIBUTION

, as it is

called today, details of which he published in 1837.

Poisson was a prolific writer and published over

300 mathematical works during his career. Unfortu-

nately, scholars at the time generally did not regard his

latter work important, and Poisson never enjoyed the

full respect of the mathematics community. It was only

after his death, on April 25, 1840, that the significance

of his many innovative ideas became apparent.

Poisson distribution See

BINOMIAL DISTRIBUTION

.

polar coordinates Invented by S

IR

I

SAAC

N

EWTON

(1642–1727) polar coordinates identify the location of

a point Pin the plane by its distance rfrom a fixed

point Oin the plane, called the origin or the pole, and

the angle θthe line segment OP makes with a fixed ray

placed at O, called the polar axis. The pole is usually

taken as the origin of a standard system of C

ARTESIAN

COORDINATES

, with the polar axis being the positive x-

axis, and the angle θmeasured in the counterclockwise

sense. The pair of numbers (r,θ) is called the polar

coordinates of P.

As an example, the point 1 unit along the x-axis,

and 1 unit along the y-axis, written (1,1) in Cartesian

coordinates, is a distance √

–

2 from the origin and makes

an angle of 45°with the x-axis, and so has polar coor-

dinates (√

–

2,45°). By adding multiples of 360°to the

angle, one can identify the same point as (√

–

2,405°),

(√

–

2,765°) or even (√

–

2, –315°), for example. This

shows that the polar coordinate representation of a

point is not unique. The polar coordinates of the origin

are not well defined, and this point is usually referred

to simply as the point with r= 0.

It is possible to convert between polar and Carte-

sian coordinates. If a point Phas Cartesian coordi-

nates (x,y) and polar coordinates (r,θ) then Plies at the

apex of a right triangle with a horizontal leg of length

xand vertical leg of length y. The length ris the

hypotenuse of the triangle and, from

TRIGONOMETRY

,

θis an angle whose sine is y/r, cosine is x/r and tangent

is y/x. Thus, with the aid of P

YTHAGORAS

’

S THEOREM

,

we have the equations:

As a check we see that the point P= (1,1) given in Carte-

sian coordinates does indeed have polar coordinates

given by r= = √

–

2 and . The

point Qwith polar coordinates (2,30°) has Cartesian

coordinates given by: x= 2 cos(30) = √

–

3 and y=

2sin(30) = 1. Thus Q= (√

–

3,1).

Polar coordinates are useful in describing equations

in mathematics that have central symmetry about the

origin. For example, in Cartesian coordinates, the

equation of a circle of radius five reads: x2+ y2= 25. In

polar coordinates, this equation reduces to (rcosθ)2+

(rsinθ)2= 25, that is, r2(cos2θ+ sin2θ) = r2= 25, or sim-

ply r= 5. The equation of an A

RCHIMEDEAN SPIRAL

is

also elementary in polar coordinates: r= aθfor some

constant a.

A

DOUBLE INTEGRAL

∫

R

∫f(x,y)dA is defined to be the

volume under the graph z= f(x,y) above a region Rin

the xy-plane. It can be computed as an iterated integral

∫∫f(x,y)dxdy with the appropriate limits of integration

inserted. To convert this integral to one expressed in

polar coordinates, one performs double integration

given by ∫∫f(rcosθ,rsinθ)r dr dθ. The appearance of the

rin the integrand is explained as follows: